Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.7%

Time: 13.5s

Alternatives: 26

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: 93.7% 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 95.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ t_2 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -0.02:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{elif}\;t\_3 \leq 10^{-17}:\\ \;\;\;\;\sin th \cdot \frac{\frac{1}{\frac{1}{ky}}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{t\_2} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (sin th)
          (/
           (sin ky)
           (hypot (sin ky) (fma kx (* -0.16666666666666666 (* kx kx)) kx)))))
        (t_2 (hypot (sin ky) (sin kx)))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_3 -1.0)
     t_1
     (if (<= t_3 -0.02)
       (*
        (* (sin ky) th)
        (sqrt
         (/
          1.0
          (fma
           0.5
           (- 1.0 (cos (* kx -2.0)))
           (fma -0.5 (cos (* ky -2.0)) 0.5)))))
       (if (<= t_3 1e-17)
         (* (sin th) (/ (/ 1.0 (/ 1.0 ky)) t_2))
         (if (<= t_3 0.995)
           (* (/ (sin ky) t_2) (fma th (* -0.16666666666666666 (* th th)) th))
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	double t_2 = hypot(sin(ky), sin(kx));
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = t_1;
	} else if (t_3 <= -0.02) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), fma(-0.5, cos((ky * -2.0)), 0.5))));
	} else if (t_3 <= 1e-17) {
		tmp = sin(th) * ((1.0 / (1.0 / ky)) / t_2);
	} else if (t_3 <= 0.995) {
		tmp = (sin(ky) / t_2) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))))
	t_2 = hypot(sin(ky), sin(kx))
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = t_1;
	elseif (t_3 <= -0.02)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(kx * -2.0))), fma(-0.5, cos(Float64(ky * -2.0)), 0.5)))));
	elseif (t_3 <= 1e-17)
		tmp = Float64(sin(th) * Float64(Float64(1.0 / Float64(1.0 / ky)) / t_2));
	elseif (t_3 <= 0.995)
		tmp = Float64(Float64(sin(ky) / t_2) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(N[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, -1.0], t$95$1, If[LessEqual[t$95$3, -0.02], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-17], N[(N[Sin[th], $MachinePrecision] * N[(N[(1.0 / N[(1.0 / ky), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$2), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

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

   1. Initial program 85.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.6

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

   7. Applied rewrites 98.6%

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

   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))))) < -0.0200000000000000004

   1. Initial program 98.9%

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 49.0%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000007e-17

   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. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. inv-pow N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in kx around inf

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in ky around 0

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

      1. lower-/.f64 98.9

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

   10. Applied rewrites 98.9%

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

   if 1.00000000000000007e-17 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 49.6

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

   7. Applied rewrites 49.6%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.02:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-17}:\\ \;\;\;\;\sin th \cdot \frac{\frac{1}{\frac{1}{ky}}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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)))