Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.2% → 99.7%

Time: 14.4s

Alternatives: 23

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.2% 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

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

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 2: 82.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \sin ky \cdot th\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_1 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.9996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_2 \cdot \sqrt{{\left(\mathsf{fma}\left(\sin ky, \sin ky, 0.5\right) - \cos \left(-2 \cdot kx\right) \cdot 0.5\right)}^{-1}}\\ \mathbf{elif}\;t\_4 \leq 0.06:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1 + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.998:\\ \;\;\;\;\frac{t\_2}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{0.5}{\sin ky} \cdot kx, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (* (sin ky) th))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_1 t_3)))))
   (if (<= t_4 -0.9996)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th))
     (if (<= t_4 -0.1)
       (*
        t_2
        (sqrt
         (pow (- (fma (sin ky) (sin ky) 0.5) (* (cos (* -2.0 kx)) 0.5)) -1.0)))
       (if (<= t_4 0.06)
         (*
          (/
           (sin ky)
           (sqrt
            (+ t_1 (* (fma (* ky ky) -0.3333333333333333 1.0) (* ky ky)))))
          (sin th))
         (if (<= t_4 0.998)
           (/ t_2 (hypot (sin ky) (sin kx)))
           (*
            (/ (sin ky) (fma (* (/ 0.5 (sin ky)) kx) kx (sin ky)))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) * th;
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_1 + t_3));
	double tmp;
	if (t_4 <= -0.9996) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	} else if (t_4 <= -0.1) {
		tmp = t_2 * sqrt(pow((fma(sin(ky), sin(ky), 0.5) - (cos((-2.0 * kx)) * 0.5)), -1.0));
	} else if (t_4 <= 0.06) {
		tmp = (sin(ky) / sqrt((t_1 + (fma((ky * ky), -0.3333333333333333, 1.0) * (ky * ky))))) * sin(th);
	} else if (t_4 <= 0.998) {
		tmp = t_2 / hypot(sin(ky), sin(kx));
	} else {
		tmp = (sin(ky) / fma(((0.5 / sin(ky)) * kx), kx, sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) * th)
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_1 + t_3)))
	tmp = 0.0
	if (t_4 <= -0.9996)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th));
	elseif (t_4 <= -0.1)
		tmp = Float64(t_2 * sqrt((Float64(fma(sin(ky), sin(ky), 0.5) - Float64(cos(Float64(-2.0 * kx)) * 0.5)) ^ -1.0)));
	elseif (t_4 <= 0.06)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(t_1 + Float64(fma(Float64(ky * ky), -0.3333333333333333, 1.0) * Float64(ky * ky))))) * sin(th));
	elseif (t_4 <= 0.998)
		tmp = Float64(t_2 / hypot(sin(ky), sin(kx)));
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(Float64(0.5 / sin(ky)) * kx), kx, sin(ky))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] * th), $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$1 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.9996], 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, -0.1], N[(t$95$2 * N[Sqrt[N[Power[N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[ky], $MachinePrecision] + 0.5), $MachinePrecision] - N[(N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.06], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.998], N[(t$95$2 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[(0.5 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * kx), $MachinePrecision] * kx + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99960000000000004

   1. Initial program 88.3%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if -0.99960000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. cos-neg-rev N/A

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

      16. lower-cos.f64 N/A

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

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

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 48.0

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

   7. Applied rewrites 48.0%

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

   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))))) < 0.059999999999999998

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.0

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \color{blue}{\left(ky \cdot ky\right)}}} \cdot \sin th \]

   8. Applied rewrites 98.0%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   if 0.059999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 50.4

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

   7. Applied rewrites 50.4%

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

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

   1. Initial program 86.3%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 93.4

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

   5. Applied rewrites 93.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\mathsf{fma}\left(\sin ky, \sin ky, 0.5\right) - \cos \left(-2 \cdot kx\right) \cdot 0.5\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.06:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.998:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{0.5}{\sin ky} \cdot kx, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \sin ky \cdot th\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.9996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_2 \cdot \sqrt{{\left(\mathsf{fma}\left(\sin ky, \sin ky, 0.5\right) - \cos \left(-2 \cdot kx\right) \cdot 0.5\right)}^{-1}}\\ \mathbf{elif}\;t\_4 \leq 0.06:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot \sin th\right) \cdot ky}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 0.998:\\ \;\;\;\;\frac{t\_2}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{0.5}{\sin ky} \cdot kx, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (* (sin ky) th))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_3)))))
   (if (<= t_4 -0.9996)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th))
     (if (<= t_4 -0.1)
       (*
        t_2
        (sqrt
         (pow (- (fma (sin ky) (sin ky) 0.5) (* (cos (* -2.0 kx)) 0.5)) -1.0)))
       (if (<= t_4 0.06)
         (/ (* (* (fma -0.16666666666666666 (* ky ky) 1.0) (sin th)) ky) t_1)
         (if (<= t_4 0.998)
           (/ t_2 t_1)
           (*
            (/ (sin ky) (fma (* (/ 0.5 (sin ky)) kx) kx (sin ky)))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = sin(ky) * th;
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_3));
	double tmp;
	if (t_4 <= -0.9996) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	} else if (t_4 <= -0.1) {
		tmp = t_2 * sqrt(pow((fma(sin(ky), sin(ky), 0.5) - (cos((-2.0 * kx)) * 0.5)), -1.0));
	} else if (t_4 <= 0.06) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * sin(th)) * ky) / t_1;
	} else if (t_4 <= 0.998) {
		tmp = t_2 / t_1;
	} else {
		tmp = (sin(ky) / fma(((0.5 / sin(ky)) * kx), kx, sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(sin(ky) * th)
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_3)))
	tmp = 0.0
	if (t_4 <= -0.9996)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th));
	elseif (t_4 <= -0.1)
		tmp = Float64(t_2 * sqrt((Float64(fma(sin(ky), sin(ky), 0.5) - Float64(cos(Float64(-2.0 * kx)) * 0.5)) ^ -1.0)));
	elseif (t_4 <= 0.06)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * sin(th)) * ky) / t_1);
	elseif (t_4 <= 0.998)
		tmp = Float64(t_2 / t_1);
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(Float64(0.5 / sin(ky)) * kx), kx, sin(ky))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] * th), $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[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.9996], 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, -0.1], N[(t$95$2 * N[Sqrt[N[Power[N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[ky], $MachinePrecision] + 0.5), $MachinePrecision] - N[(N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.06], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 0.998], N[(t$95$2 / t$95$1), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[(0.5 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * kx), $MachinePrecision] * kx + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99960000000000004

   1. Initial program 88.3%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if -0.99960000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. cos-neg-rev N/A

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

      16. lower-cos.f64 N/A

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

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

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 48.0

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

   7. Applied rewrites 48.0%

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

   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))))) < 0.059999999999999998

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 97.8

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

   7. Applied rewrites 97.8%

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

   if 0.059999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 50.4

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

   7. Applied rewrites 50.4%

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

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

   1. Initial program 86.3%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 93.4

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

   5. Applied rewrites 93.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\mathsf{fma}\left(\sin ky, \sin ky, 0.5\right) - \cos \left(-2 \cdot kx\right) \cdot 0.5\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.06:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.998:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{0.5}{\sin ky} \cdot kx, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky \cdot th}{t\_1}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.06:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot \sin th\right) \cdot ky}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 0.998:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{0.5}{\sin ky} \cdot kx, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (/ (* (sin ky) th) t_1))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_3)))))
   (if (<= t_4 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th))
     (if (<= t_4 -0.1)
       t_2
       (if (<= t_4 0.06)
         (/ (* (* (fma -0.16666666666666666 (* ky ky) 1.0) (sin th)) ky) t_1)
         (if (<= t_4 0.998)
           t_2
           (*
            (/ (sin ky) (fma (* (/ 0.5 (sin ky)) kx) kx (sin ky)))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = (sin(ky) * th) / t_1;
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_3));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	} else if (t_4 <= -0.1) {
		tmp = t_2;
	} else if (t_4 <= 0.06) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * sin(th)) * ky) / t_1;
	} else if (t_4 <= 0.998) {
		tmp = t_2;
	} else {
		tmp = (sin(ky) / fma(((0.5 / sin(ky)) * kx), kx, sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(Float64(sin(ky) * th) / t_1)
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_3)))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th));
	elseif (t_4 <= -0.1)
		tmp = t_2;
	elseif (t_4 <= 0.06)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * sin(th)) * ky) / t_1);
	elseif (t_4 <= 0.998)
		tmp = t_2;
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(Float64(0.5 / sin(ky)) * kx), kx, sin(ky))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $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[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.1], t$95$2, If[LessEqual[t$95$4, 0.06], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 0.998], t$95$2, N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[(0.5 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * kx), $MachinePrecision] * kx + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 87.6%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 0.059999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 48.7

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

   7. Applied rewrites 48.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.059999999999999998

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 97.8

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

   7. Applied rewrites 97.8%

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

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

   1. Initial program 86.3%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 93.4

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

   5. Applied rewrites 93.4%

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

Alternative 5: 81.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky \cdot th}{t\_1}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.06:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot \sin th\right) \cdot ky}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 0.998:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (/ (* (sin ky) th) t_1))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_3)))))
   (if (<= t_4 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th))
     (if (<= t_4 -0.1)
       t_2
       (if (<= t_4 0.06)
         (/ (* (* (fma -0.16666666666666666 (* ky ky) 1.0) (sin th)) ky) t_1)
         (if (<= t_4 0.998) t_2 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = (sin(ky) * th) / t_1;
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_3));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	} else if (t_4 <= -0.1) {
		tmp = t_2;
	} else if (t_4 <= 0.06) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * sin(th)) * ky) / t_1;
	} else if (t_4 <= 0.998) {
		tmp = t_2;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(Float64(sin(ky) * th) / t_1)
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_3)))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th));
	elseif (t_4 <= -0.1)
		tmp = t_2;
	elseif (t_4 <= 0.06)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * sin(th)) * ky) / t_1);
	elseif (t_4 <= 0.998)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $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[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.1], t$95$2, If[LessEqual[t$95$4, 0.06], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 0.998], t$95$2, N[Sin[th], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 87.6%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 0.059999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 48.7

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

   7. Applied rewrites 48.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.059999999999999998

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 97.8

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

   7. Applied rewrites 97.8%

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

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

   1. Initial program 86.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 93.2

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

   5. Applied rewrites 93.2%

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

Alternative 6: 81.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky \cdot th}{t\_1}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\left(\sqrt{{t\_3}^{-1}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.06:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot \sin th\right) \cdot ky}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 0.998:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (/ (* (sin ky) th) t_1))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_3)))))
   (if (<= t_4 -1.0)
     (* (* (sqrt (pow t_3 -1.0)) (sin ky)) (sin th))
     (if (<= t_4 -0.1)
       t_2
       (if (<= t_4 0.06)
         (/ (* (* (fma -0.16666666666666666 (* ky ky) 1.0) (sin th)) ky) t_1)
         (if (<= t_4 0.998) t_2 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = (sin(ky) * th) / t_1;
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_3));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = (sqrt(pow(t_3, -1.0)) * sin(ky)) * sin(th);
	} else if (t_4 <= -0.1) {
		tmp = t_2;
	} else if (t_4 <= 0.06) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * sin(th)) * ky) / t_1;
	} else if (t_4 <= 0.998) {
		tmp = t_2;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(Float64(sin(ky) * th) / t_1)
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_3)))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = Float64(Float64(sqrt((t_3 ^ -1.0)) * sin(ky)) * sin(th));
	elseif (t_4 <= -0.1)
		tmp = t_2;
	elseif (t_4 <= 0.06)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * sin(th)) * ky) / t_1);
	elseif (t_4 <= 0.998)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $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[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], N[(N[(N[Sqrt[N[Power[t$95$3, -1.0], $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.1], t$95$2, If[LessEqual[t$95$4, 0.06], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 0.998], t$95$2, N[Sin[th], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 87.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 87.6

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 87.4%

       \[\leadsto \left(\sqrt{\frac{1}{{\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 0.059999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 48.7

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

   7. Applied rewrites 48.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.059999999999999998

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 97.8

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

   7. Applied rewrites 97.8%

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

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

   1. Initial program 86.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\left(\sqrt{{\left({\sin ky}^{2}\right)}^{-1}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.06:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.998:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky \cdot th}{t\_1}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\left(\sqrt{{t\_3}^{-1}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.07:\\ \;\;\;\;\frac{\sin th \cdot ky}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 0.998:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (/ (* (sin ky) th) t_1))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_3)))))
   (if (<= t_4 -1.0)
     (* (* (sqrt (pow t_3 -1.0)) (sin ky)) (sin th))
     (if (<= t_4 -0.1)
       t_2
       (if (<= t_4 0.07)
         (/ (* (sin th) ky) t_1)
         (if (<= t_4 0.998) t_2 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = (sin(ky) * th) / t_1;
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_3));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = (sqrt(pow(t_3, -1.0)) * sin(ky)) * sin(th);
	} else if (t_4 <= -0.1) {
		tmp = t_2;
	} else if (t_4 <= 0.07) {
		tmp = (sin(th) * ky) / t_1;
	} else if (t_4 <= 0.998) {
		tmp = t_2;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double t_2 = (Math.sin(ky) * th) / t_1;
	double t_3 = Math.pow(Math.sin(ky), 2.0);
	double t_4 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_3));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = (Math.sqrt(Math.pow(t_3, -1.0)) * Math.sin(ky)) * Math.sin(th);
	} else if (t_4 <= -0.1) {
		tmp = t_2;
	} else if (t_4 <= 0.07) {
		tmp = (Math.sin(th) * ky) / t_1;
	} else if (t_4 <= 0.998) {
		tmp = t_2;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	t_2 = (math.sin(ky) * th) / t_1
	t_3 = math.pow(math.sin(ky), 2.0)
	t_4 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_3))
	tmp = 0
	if t_4 <= -1.0:
		tmp = (math.sqrt(math.pow(t_3, -1.0)) * math.sin(ky)) * math.sin(th)
	elif t_4 <= -0.1:
		tmp = t_2
	elif t_4 <= 0.07:
		tmp = (math.sin(th) * ky) / t_1
	elif t_4 <= 0.998:
		tmp = t_2
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(Float64(sin(ky) * th) / t_1)
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_3)))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = Float64(Float64(sqrt((t_3 ^ -1.0)) * sin(ky)) * sin(th));
	elseif (t_4 <= -0.1)
		tmp = t_2;
	elseif (t_4 <= 0.07)
		tmp = Float64(Float64(sin(th) * ky) / t_1);
	elseif (t_4 <= 0.998)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	t_2 = (sin(ky) * th) / t_1;
	t_3 = sin(ky) ^ 2.0;
	t_4 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_3));
	tmp = 0.0;
	if (t_4 <= -1.0)
		tmp = (sqrt((t_3 ^ -1.0)) * sin(ky)) * sin(th);
	elseif (t_4 <= -0.1)
		tmp = t_2;
	elseif (t_4 <= 0.07)
		tmp = (sin(th) * ky) / t_1;
	elseif (t_4 <= 0.998)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $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[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], N[(N[(N[Sqrt[N[Power[t$95$3, -1.0], $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.1], t$95$2, If[LessEqual[t$95$4, 0.07], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 0.998], t$95$2, N[Sin[th], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 87.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 87.6

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 87.4%

       \[\leadsto \left(\sqrt{\frac{1}{{\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 0.070000000000000007 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 48.7

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

   7. Applied rewrites 48.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.070000000000000007

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 97.3

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

   7. Applied rewrites 97.3%

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

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

   1. Initial program 86.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\left(\sqrt{{\left({\sin ky}^{2}\right)}^{-1}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.07:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.998:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky \cdot th}{t\_1}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.07:\\ \;\;\;\;\frac{\sin th \cdot ky}{t\_1}\\ \mathbf{elif}\;t\_3 \leq 0.998:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (/ (* (sin ky) th) t_1))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_3 -0.1)
     t_2
     (if (<= t_3 0.07)
       (/ (* (sin th) ky) t_1)
       (if (<= t_3 0.998) t_2 (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = (sin(ky) * th) / t_1;
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.1) {
		tmp = t_2;
	} else if (t_3 <= 0.07) {
		tmp = (sin(th) * ky) / t_1;
	} else if (t_3 <= 0.998) {
		tmp = t_2;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double t_2 = (Math.sin(ky) * th) / t_1;
	double t_3 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.1) {
		tmp = t_2;
	} else if (t_3 <= 0.07) {
		tmp = (Math.sin(th) * ky) / t_1;
	} else if (t_3 <= 0.998) {
		tmp = t_2;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	t_2 = (math.sin(ky) * th) / t_1
	t_3 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_3 <= -0.1:
		tmp = t_2
	elif t_3 <= 0.07:
		tmp = (math.sin(th) * ky) / t_1
	elif t_3 <= 0.998:
		tmp = t_2
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(Float64(sin(ky) * th) / t_1)
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.1)
		tmp = t_2;
	elseif (t_3 <= 0.07)
		tmp = Float64(Float64(sin(th) * ky) / t_1);
	elseif (t_3 <= 0.998)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	t_2 = (sin(ky) * th) / t_1;
	t_3 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -0.1)
		tmp = t_2;
	elseif (t_3 <= 0.07)
		tmp = (sin(th) * ky) / t_1;
	elseif (t_3 <= 0.998)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.1], t$95$2, If[LessEqual[t$95$3, 0.07], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.998], t$95$2, 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 or 0.070000000000000007 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 96.4

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 42.8

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

   7. Applied rewrites 42.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.070000000000000007

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 97.3

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

   7. Applied rewrites 97.3%

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

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

   1. Initial program 86.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 93.2

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

   5. Applied rewrites 93.2%

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

Alternative 9: 61.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.07:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.998:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* (sin ky) th) (hypot (sin ky) (sin kx))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.1)
     t_1
     (if (<= t_2 0.07)
       (* (sin ky) (/ (sin th) (sin kx)))
       (if (<= t_2 0.998) t_1 (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = t_1;
	} else if (t_2 <= 0.07) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (t_2 <= 0.998) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) * th) / Math.hypot(Math.sin(ky), Math.sin(kx));
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = t_1;
	} else if (t_2 <= 0.07) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (t_2 <= 0.998) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) * th) / math.hypot(math.sin(ky), math.sin(kx))
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= -0.1:
		tmp = t_1
	elif t_2 <= 0.07:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif t_2 <= 0.998:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.1)
		tmp = t_1;
	elseif (t_2 <= 0.07)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (t_2 <= 0.998)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.1)
		tmp = t_1;
	elseif (t_2 <= 0.07)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (t_2 <= 0.998)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.1], t$95$1, If[LessEqual[t$95$2, 0.07], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.998], t$95$1, 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 or 0.070000000000000007 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 96.4

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 42.8

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

   7. Applied rewrites 42.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.070000000000000007

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 70.7

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

   7. Applied rewrites 70.7%

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

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

   1. Initial program 86.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 93.2

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

   5. Applied rewrites 93.2%

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

Alternative 10: 59.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.07:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.998:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (/
           (sin ky)
           (/
            (sqrt
             (fma
              (- 1.0 (cos (* 2.0 ky)))
              2.0
              (* 2.0 (- 1.0 (cos (* 2.0 kx))))))
            2.0))
          (* (fma (* th th) -0.16666666666666666 1.0) th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.1)
     t_1
     (if (<= t_2 0.07)
       (* (sin ky) (/ (sin th) (sin kx)))
       (if (<= t_2 0.998) t_1 (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / (sqrt(fma((1.0 - cos((2.0 * ky))), 2.0, (2.0 * (1.0 - cos((2.0 * kx)))))) / 2.0)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = t_1;
	} else if (t_2 <= 0.07) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (t_2 <= 0.998) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 2.0, Float64(2.0 * Float64(1.0 - cos(Float64(2.0 * kx)))))) / 2.0)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.1)
		tmp = t_1;
	elseif (t_2 <= 0.07)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (t_2 <= 0.998)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(2.0 * N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.1], t$95$1, If[LessEqual[t$95$2, 0.07], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.998], t$95$1, 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 or 0.070000000000000007 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 94.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.7

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

   7. Applied rewrites 45.7%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. frac-add N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   9. Applied rewrites 37.7%

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

   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))))) < 0.070000000000000007

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 70.7

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

   7. Applied rewrites 70.7%

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

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

   1. Initial program 86.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 93.2

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

   5. Applied rewrites 93.2%

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

Alternative 11: 46.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.1)
   (* (sin ky) (/ (sin th) (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.1) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.1d0) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.1) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.1:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.1)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.1)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.1], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 38.6

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

   5. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 38.6

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

   7. Applied rewrites 38.6%

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

   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)))))

   1. Initial program 91.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.1

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

   5. Applied rewrites 67.1%

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

Alternative 12: 45.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.06:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.06)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.06) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.06d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.06) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.06:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.06)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.06)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.06], 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))))) < 0.059999999999999998

   1. Initial program 96.2%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 37.1

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

   5. Applied rewrites 37.1%

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

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

   1. Initial program 91.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 66.6

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

   5. Applied rewrites 66.6%

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

Alternative 13: 44.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.06:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.06)
   (/ (* (sin th) ky) (sin kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.06) {
		tmp = (sin(th) * ky) / sin(kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.06d0) then
        tmp = (sin(th) * ky) / sin(kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.06) {
		tmp = (Math.sin(th) * ky) / Math.sin(kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.06:
		tmp = (math.sin(th) * ky) / math.sin(kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.06)
		tmp = Float64(Float64(sin(th) * ky) / sin(kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.06)
		tmp = (sin(th) * ky) / sin(kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.06], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 ky around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 36.6

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

   7. Applied rewrites 36.6%

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

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

   1. Initial program 91.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 66.6

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

   5. Applied rewrites 66.6%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.06:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 15.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 10^{-322}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      1e-322)
   (* (* (* -0.16666666666666666 th) th) th)
   (*
    (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(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 1e-322) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = 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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 1e-322)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th);
	else
		tmp = Float64(fma(Float64(Float64(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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 1e-322], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 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)) < 9.88131e-323

   1. Initial program 94.7%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 26.9

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

   5. Applied rewrites 26.9%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 13.5%

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

   9. Taylor expanded in th around inf

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

   11. Applied rewrites 13.4%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 13.4%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 9.88131e-323 < (*.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 94.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.7%

      \[\leadsto \mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot \color{blue}{th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 15.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 10^{-322}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      1e-322)
   (* (* (* -0.16666666666666666 th) th) 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(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 1e-322) {
		tmp = ((-0.16666666666666666 * th) * th) * 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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 1e-322)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * 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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 1e-322], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $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)) < 9.88131e-323

   1. Initial program 94.7%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 26.9

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

   5. Applied rewrites 26.9%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 13.5%

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

   9. Taylor expanded in th around inf

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

   11. Applied rewrites 13.4%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 13.4%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 9.88131e-323 < (*.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 94.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 16.8%

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

   10. Applied rewrites 16.8%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \cdot th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 36.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-5)
   (/ (* (sin th) ky) kx)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5e-5) {
		tmp = (sin(th) * ky) / kx;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 5d-5) then
        tmp = (sin(th) * ky) / kx
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 5e-5) {
		tmp = (Math.sin(th) * ky) / kx;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 5e-5:
		tmp = (math.sin(th) * ky) / kx
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-5)
		tmp = Float64(Float64(sin(th) * ky) / kx);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-5)
		tmp = (sin(th) * ky) / kx;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.00000000000000024e-5

   1. Initial program 96.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 83.3

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 36.4

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 21.1%

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

   if 5.00000000000000024e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 66.0

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

   5. Applied rewrites 66.0%

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

Alternative 17: 32.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1.06 \cdot 10^{-33}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
      1.06e-33)
   (* (* (* -0.16666666666666666 th) th) th)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 1.06e-33) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 1.06d-33) then
        tmp = (((-0.16666666666666666d0) * th) * th) * th
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 1.06e-33) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 1.06e-33:
		tmp = ((-0.16666666666666666 * th) * th) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1.06e-33)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1.06e-33)
		tmp = ((-0.16666666666666666 * th) * th) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.06e-33], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.0599999999999999e-33

   1. Initial program 96.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 3.6

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 3.2%

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

   9. Taylor expanded in th around inf

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

   11. Applied rewrites 11.5%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 11.5%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 1.0599999999999999e-33 < (/.f64 (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. Taylor expanded in kx around 0

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

      1. lower-sin.f64 65.4

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

   5. Applied rewrites 65.4%

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

Alternative 18: 46.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-151}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right) + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-151)
   (* (sin ky) (/ (sin th) (sin kx)))
   (if (<= (sin ky) 5e-5)
     (*
      (/
       (sin ky)
       (sqrt
        (+
         (- 0.5 (* (cos (* 2.0 kx)) 0.5))
         (* (fma (* ky ky) -0.3333333333333333 1.0) (* ky ky)))))
      (sin th))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-151) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (sin(ky) <= 5e-5) {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * kx)) * 0.5)) + (fma((ky * ky), -0.3333333333333333, 1.0) * (ky * ky))))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-151)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (sin(ky) <= 5e-5)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5)) + Float64(fma(Float64(ky * ky), -0.3333333333333333, 1.0) * Float64(ky * ky))))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-151], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-5], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < 9.9999999999999994e-152

   1. Initial program 90.8%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 34.1

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

   5. Applied rewrites 34.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 34.1

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

   7. Applied rewrites 34.1%

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

   if 9.9999999999999994e-152 < (sin.f64 ky) < 5.00000000000000024e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.9

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \color{blue}{\left(ky \cdot ky\right)}}} \cdot \sin th \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 96.8

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   10. Applied rewrites 96.8%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   if 5.00000000000000024e-5 < (sin.f64 ky)

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 60.1

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

   5. Applied rewrites 60.1%

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

Alternative 19: 70.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.00156:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}} \cdot \sin ky\right) \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.00156)
   (/
    (* (* (fma -0.16666666666666666 (* ky ky) 1.0) (sin th)) ky)
    (hypot (sin ky) (sin kx)))
   (*
    (*
     (sqrt
      (pow (/ (+ (- 1.0 (cos (* 2.0 ky))) (- 1.0 (cos (* 2.0 kx)))) 2.0) -1.0))
     (sin ky))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.00156) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * sin(th)) * ky) / hypot(sin(ky), sin(kx));
	} else {
		tmp = (sqrt(pow((((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx)))) / 2.0), -1.0)) * sin(ky)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.00156)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * sin(th)) * ky) / hypot(sin(ky), sin(kx)));
	else
		tmp = Float64(Float64(sqrt((Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) + Float64(1.0 - cos(Float64(2.0 * kx)))) / 2.0) ^ -1.0)) * sin(ky)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.00156], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[Power[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], -1.0], $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.00155999999999999997

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 94.8

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

   4. Applied rewrites 94.8%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\sin th + \color{blue}{{ky}^{2} \cdot \left(\sin th \cdot \frac{-1}{6}\right)}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\sin th + {ky}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin th\right)}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sin th + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sin th\right)\right) \cdot ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\sin th + \color{blue}{\left({ky}^{2} \cdot \frac{-1}{6}\right) \cdot \sin th}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\sin th + \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2}\right)} \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot \sin th\right)} \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)} \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \sin th\right)} \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      15. lower-sin.f64 60.5

          \[\leadsto \frac{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot \color{blue}{\sin th}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   7. Applied rewrites 60.5%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot \sin th\right) \cdot ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   if 0.00155999999999999997 < 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

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 99.6

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{\left(\frac{1}{2} + {\sin ky}^{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\left(\frac{1}{2} + {\sin ky}^{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\left(\frac{1}{2} + {\sin ky}^{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5\right) - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot \sin ky\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 98.9%

      \[\leadsto \left(\sqrt{\frac{1}{\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}}} \cdot \sin ky\right) \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.00156:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}} \cdot \sin ky\right) \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 70.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.00156:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right) + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.00156)
   (/
    (* (* (fma -0.16666666666666666 (* ky ky) 1.0) (sin th)) ky)
    (hypot (sin ky) (sin kx)))
   (*
    (/
     (sin ky)
     (sqrt
      (+ (- 0.5 (* (cos (* 2.0 kx)) 0.5)) (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.00156) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * sin(th)) * ky) / hypot(sin(ky), sin(kx));
	} else {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * kx)) * 0.5)) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.00156)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * sin(th)) * ky) / hypot(sin(ky), sin(kx)));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5)) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.00156], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $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]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.00155999999999999997

   1. Initial program 92.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lower-*.f64 90.3

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \]

      14. lower-hypot.f64 94.8

          \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   4. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(\sin th + \frac{-1}{6} \cdot \left({ky}^{2} \cdot \sin th\right)\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin th + \frac{-1}{6} \cdot \left({ky}^{2} \cdot \sin th\right)\right) \cdot ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\sin th + \color{blue}{\left({ky}^{2} \cdot \sin th\right) \cdot \frac{-1}{6}}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\sin th + \color{blue}{{ky}^{2} \cdot \left(\sin th \cdot \frac{-1}{6}\right)}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\sin th + {ky}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin th\right)}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sin th + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sin th\right)\right) \cdot ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\sin th + \color{blue}{\left({ky}^{2} \cdot \frac{-1}{6}\right) \cdot \sin th}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\sin th + \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2}\right)} \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot \sin th\right)} \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)} \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \sin th\right)} \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      15. lower-sin.f64 60.5

          \[\leadsto \frac{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot \color{blue}{\sin th}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   7. Applied rewrites 60.5%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot \sin th\right) \cdot ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   if 0.00155999999999999997 < 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

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 99.6

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      9. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \left(\frac{1}{2} - \cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 99.0

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right) + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   6. Applied rewrites 99.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right) + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 39.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.26 \cdot 10^{-144}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)}^{-1}} \cdot \sin ky\right) \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.26e-144)
   (sin th)
   (if (<= kx 1.4e-6)
     (*
      (/
       (sin ky)
       (sqrt
        (+ (* kx kx) (* (fma (* ky ky) -0.3333333333333333 1.0) (* ky ky)))))
      (sin th))
     (*
      (* (sqrt (pow (- 0.5 (* (cos (* 2.0 kx)) 0.5)) -1.0)) (sin ky))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.26e-144) {
		tmp = sin(th);
	} else if (kx <= 1.4e-6) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (fma((ky * ky), -0.3333333333333333, 1.0) * (ky * ky))))) * sin(th);
	} else {
		tmp = (sqrt(pow((0.5 - (cos((2.0 * kx)) * 0.5)), -1.0)) * sin(ky)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.26e-144)
		tmp = sin(th);
	elseif (kx <= 1.4e-6)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(fma(Float64(ky * ky), -0.3333333333333333, 1.0) * Float64(ky * ky))))) * sin(th));
	else
		tmp = Float64(Float64(sqrt((Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5)) ^ -1.0)) * sin(ky)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.26e-144], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 1.4e-6], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[Power[N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 1.26e-144

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 30.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\sin th} \]

   if 1.26e-144 < kx < 1.39999999999999994e-6

   1. Initial program 99.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   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 56.5

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   5. Applied rewrites 56.5%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 + \frac{-1}{3} \cdot {ky}^{2}\right) \cdot {ky}^{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 + \frac{-1}{3} \cdot {ky}^{2}\right) \cdot {ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(\frac{-1}{3} \cdot {ky}^{2} + 1\right)} \cdot {ky}^{2}}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(\color{blue}{{ky}^{2} \cdot \frac{-1}{3}} + 1\right) \cdot {ky}^{2}}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{3}, 1\right)} \cdot {ky}^{2}}} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{3}, 1\right) \cdot {ky}^{2}}} \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{3}, 1\right) \cdot {ky}^{2}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right) \cdot \color{blue}{\left(ky \cdot ky\right)}}} \cdot \sin th \]

      9. lower-*.f64 53.9

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \color{blue}{\left(ky \cdot ky\right)}}} \cdot \sin th \]

   8. Applied rewrites 53.9%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   9. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       2. lower-*.f64 53.9

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   11. Applied rewrites 53.9%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   if 1.39999999999999994e-6 < 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-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 99.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{\left(\frac{1}{2} + {\sin ky}^{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\left(\frac{1}{2} + {\sin ky}^{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\left(\frac{1}{2} + {\sin ky}^{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5\right) - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot \sin ky\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 57.4%

       \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.26 \cdot 10^{-144}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)}^{-1}} \cdot \sin ky\right) \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 39.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.26 \cdot 10^{-144}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.26e-144)
   (sin th)
   (if (<= kx 1.4e-6)
     (*
      (/
       (sin ky)
       (sqrt
        (+ (* kx kx) (* (fma (* ky ky) -0.3333333333333333 1.0) (* ky ky)))))
      (sin th))
     (* (/ (sin ky) (sqrt (- 0.5 (* (cos (* -2.0 kx)) 0.5)))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.26e-144) {
		tmp = sin(th);
	} else if (kx <= 1.4e-6) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (fma((ky * ky), -0.3333333333333333, 1.0) * (ky * ky))))) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt((0.5 - (cos((-2.0 * kx)) * 0.5)))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.26e-144)
		tmp = sin(th);
	elseif (kx <= 1.4e-6)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(fma(Float64(ky * ky), -0.3333333333333333, 1.0) * Float64(ky * ky))))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(-2.0 * kx)) * 0.5)))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.26e-144], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 1.4e-6], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 1.26e-144

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 30.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\sin th} \]

   if 1.26e-144 < kx < 1.39999999999999994e-6

   1. Initial program 99.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   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 56.5

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   5. Applied rewrites 56.5%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 + \frac{-1}{3} \cdot {ky}^{2}\right) \cdot {ky}^{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 + \frac{-1}{3} \cdot {ky}^{2}\right) \cdot {ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(\frac{-1}{3} \cdot {ky}^{2} + 1\right)} \cdot {ky}^{2}}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(\color{blue}{{ky}^{2} \cdot \frac{-1}{3}} + 1\right) \cdot {ky}^{2}}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{3}, 1\right)} \cdot {ky}^{2}}} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{3}, 1\right) \cdot {ky}^{2}}} \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{3}, 1\right) \cdot {ky}^{2}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right) \cdot \color{blue}{\left(ky \cdot ky\right)}}} \cdot \sin th \]

      9. lower-*.f64 53.9

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \color{blue}{\left(ky \cdot ky\right)}}} \cdot \sin th \]

   8. Applied rewrites 53.9%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   9. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       2. lower-*.f64 53.9

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   11. Applied rewrites 53.9%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   if 1.39999999999999994e-6 < 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-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 99.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      5. cos-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      9. metadata-eval 57.3

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot \sin th \]

   7. Applied rewrites 57.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 11.4% accurate, 39.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (* (* -0.16666666666666666 th) th) th))

C

double code(double kx, double ky, double th) {
	return ((-0.16666666666666666 * th) * th) * 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 = (((-0.16666666666666666d0) * th) * th) * th
end function

Java

public static double code(double kx, double ky, double th) {
	return ((-0.16666666666666666 * th) * th) * th;
}

Python

def code(kx, ky, th):
	return ((-0.16666666666666666 * th) * th) * th

Julia

function code(kx, ky, th)
	return Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = ((-0.16666666666666666 * th) * th) * th;
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $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. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 25.8

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 25.8%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 15.1%

   \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

9. Taylor expanded in th around inf

   \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
10. Step-by-step derivation

11. Applied rewrites 8.7%

    \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
12. Step-by-step derivation

13. Applied rewrites 8.7%

    \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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)))