Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.4% → 99.7%

Time: 13.6s

Alternatives: 27

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: 93.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

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 92.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

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: 80.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\right)\\ \mathbf{if}\;t\_2 \leq -0.999:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right) + kx \cdot kx}}\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(t\_1, \sin kx\right)}\\ \mathbf{elif}\;t\_2 \leq 0.99:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(t\_1, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fma ky (* -0.16666666666666666 (* ky ky)) ky))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3
         (*
          (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_2 -0.999)
     (*
      (sin th)
      (/ (sin ky) (sqrt (+ (fma (cos (+ ky ky)) -0.5 0.5) (* kx kx)))))
     (if (<= t_2 -0.2)
       t_3
       (if (<= t_2 0.02)
         (* (sin th) (/ (sin ky) (hypot t_1 (sin kx))))
         (if (<= t_2 0.99)
           t_3
           (if (<= t_2 1.0)
             (sin th)
             (*
              (sin th)
              (/
               (sin ky)
               (hypot
                t_1
                (fma kx (* -0.16666666666666666 (* kx kx)) kx)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma(ky, (-0.16666666666666666 * (ky * ky)), ky);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = sin(ky) * (th * sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), fma(-0.5, cos((ky * -2.0)), 0.5)))));
	double tmp;
	if (t_2 <= -0.999) {
		tmp = sin(th) * (sin(ky) / sqrt((fma(cos((ky + ky)), -0.5, 0.5) + (kx * kx))));
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 0.02) {
		tmp = sin(th) * (sin(ky) / hypot(t_1, sin(kx)));
	} else if (t_2 <= 0.99) {
		tmp = t_3;
	} else if (t_2 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (sin(ky) / hypot(t_1, fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(sin(ky) * Float64(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))))))
	tmp = 0.0
	if (t_2 <= -0.999)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(fma(cos(Float64(ky + ky)), -0.5, 0.5) + Float64(kx * kx)))));
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 0.02)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(t_1, sin(kx))));
	elseif (t_2 <= 0.99)
		tmp = t_3;
	elseif (t_2 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(t_1, fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] * N[(th * 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]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.999], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], t$95$3, If[LessEqual[t$95$2, 0.02], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.99], t$95$3, If[LessEqual[t$95$2, 1.0], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1 ^ 2 + N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 85.5%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 84.5

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

   5. Applied rewrites 84.5%

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

   7. Applied rewrites 62.6%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.98999999999999999

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

   7. Applied rewrites 54.2%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 96.7

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

   7. Applied rewrites 96.7%

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

   if 0.98999999999999999 < (/.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 99.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 96.4

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

   5. Applied rewrites 96.4%

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

   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)))))

   1. Initial program 3.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   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 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 99.7

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

   10. Applied rewrites 99.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.999:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right) + kx \cdot kx}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right), \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.99:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right), \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(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)))