Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.3% → 99.7%

Time: 10.5s

Alternatives: 16

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

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

Alternative 2: 77.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + t\_1}}\\ t_3 := \frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right)}{th}}\\ \mathbf{if}\;t\_2 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot ky\right), -0.5, 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.15:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) t_1))))
        (t_3
         (/
          (sin ky)
          (/
           (*
            (hypot (sin kx) (sin ky))
            (fma (* th th) 0.16666666666666666 1.0))
           th))))
   (if (<= t_2 -0.996)
     (* (/ (sin ky) (sqrt (fma (cos (* 2.0 ky)) -0.5 0.5))) (sin th))
     (if (<= t_2 -0.05)
       t_3
       (if (<= t_2 0.15)
         (* (/ (sin ky) (sqrt (+ (* ky ky) t_1))) (sin th))
         (if (<= t_2 0.995) t_3 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(ky), 2.0) + t_1));
	double t_3 = sin(ky) / ((hypot(sin(kx), sin(ky)) * fma((th * th), 0.16666666666666666, 1.0)) / th);
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (sin(ky) / sqrt(fma(cos((2.0 * ky)), -0.5, 0.5))) * sin(th);
	} else if (t_2 <= -0.05) {
		tmp = t_3;
	} else if (t_2 <= 0.15) {
		tmp = (sin(ky) / sqrt(((ky * ky) + t_1))) * sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + t_1)))
	t_3 = Float64(sin(ky) / Float64(Float64(hypot(sin(kx), sin(ky)) * fma(Float64(th * th), 0.16666666666666666, 1.0)) / th))
	tmp = 0.0
	if (t_2 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(cos(Float64(2.0 * ky)), -0.5, 0.5))) * sin(th));
	elseif (t_2 <= -0.05)
		tmp = t_3;
	elseif (t_2 <= 0.15)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + t_1))) * sin(th));
	elseif (t_2 <= 0.995)
		tmp = t_3;
	else
		tmp = 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] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(th * th), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.05], t$95$3, If[LessEqual[t$95$2, 0.15], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.995], t$95$3, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 87.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 61.1

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 59.5

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

   7. Applied rewrites 59.5%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003 or 0.149999999999999994 < (/.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 99.2%

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

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

      7. lower-/.f64 99.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-/.f64 N/A

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

   9. Applied rewrites 49.9%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.149999999999999994

   1. Initial program 99.2%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   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 98.1

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

   5. Applied rewrites 98.1%

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

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

   1. Initial program 87.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot ky\right), -0.5, 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.15:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + {\sin kx}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + t\_1}}\\ t_3 := \frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{if}\;t\_2 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot ky\right), -0.5, 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.15:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) t_1))))
        (t_3 (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))))
   (if (<= t_2 -0.996)
     (* (/ (sin ky) (sqrt (fma (cos (* 2.0 ky)) -0.5 0.5))) (sin th))
     (if (<= t_2 -0.05)
       t_3
       (if (<= t_2 0.15)
         (* (/ (sin ky) (sqrt (+ (* ky ky) t_1))) (sin th))
         (if (<= t_2 0.995) t_3 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(ky), 2.0) + t_1));
	double t_3 = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (sin(ky) / sqrt(fma(cos((2.0 * ky)), -0.5, 0.5))) * sin(th);
	} else if (t_2 <= -0.05) {
		tmp = t_3;
	} else if (t_2 <= 0.15) {
		tmp = (sin(ky) / sqrt(((ky * ky) + t_1))) * sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + t_1)))
	t_3 = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th))
	tmp = 0.0
	if (t_2 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(cos(Float64(2.0 * ky)), -0.5, 0.5))) * sin(th));
	elseif (t_2 <= -0.05)
		tmp = t_3;
	elseif (t_2 <= 0.15)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + t_1))) * sin(th));
	elseif (t_2 <= 0.995)
		tmp = t_3;
	else
		tmp = 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] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.05], t$95$3, If[LessEqual[t$95$2, 0.15], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.995], t$95$3, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 87.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 61.1

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 59.5

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

   7. Applied rewrites 59.5%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003 or 0.149999999999999994 < (/.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 99.2%

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

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

      7. lower-/.f64 99.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 48.8

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

   9. Applied rewrites 48.8%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.149999999999999994

   1. Initial program 99.2%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   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 98.1

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

   5. Applied rewrites 98.1%

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

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

   1. Initial program 87.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot ky\right), -0.5, 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.15:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + {\sin kx}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + t\_1}}\\ t_3 := \frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{if}\;t\_2 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot ky\right), -0.5, 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.15:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{t\_1}}{\sin ky}}\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) t_1))))
        (t_3 (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))))
   (if (<= t_2 -0.996)
     (* (/ (sin ky) (sqrt (fma (cos (* 2.0 ky)) -0.5 0.5))) (sin th))
     (if (<= t_2 -0.05)
       t_3
       (if (<= t_2 0.15)
         (/ (sin th) (/ (sqrt t_1) (sin ky)))
         (if (<= t_2 0.995) t_3 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(ky), 2.0) + t_1));
	double t_3 = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (sin(ky) / sqrt(fma(cos((2.0 * ky)), -0.5, 0.5))) * sin(th);
	} else if (t_2 <= -0.05) {
		tmp = t_3;
	} else if (t_2 <= 0.15) {
		tmp = sin(th) / (sqrt(t_1) / sin(ky));
	} else if (t_2 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + t_1)))
	t_3 = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th))
	tmp = 0.0
	if (t_2 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(cos(Float64(2.0 * ky)), -0.5, 0.5))) * sin(th));
	elseif (t_2 <= -0.05)
		tmp = t_3;
	elseif (t_2 <= 0.15)
		tmp = Float64(sin(th) / Float64(sqrt(t_1) / sin(ky)));
	elseif (t_2 <= 0.995)
		tmp = t_3;
	else
		tmp = 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] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.05], t$95$3, If[LessEqual[t$95$2, 0.15], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.995], t$95$3, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 87.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 61.1

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 59.5

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

   7. Applied rewrites 59.5%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003 or 0.149999999999999994 < (/.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 99.2%

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

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

      7. lower-/.f64 99.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 48.8

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

   9. Applied rewrites 48.8%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.149999999999999994

   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 63.8

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

   7. Applied rewrites 63.8%

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

   9. Applied rewrites 98.1%

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

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

   1. Initial program 87.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot ky\right), -0.5, 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.15:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_2 := \frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{if}\;t\_1 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot ky\right), -0.5, 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_2 (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))))
   (if (<= t_1 -0.996)
     (* (/ (sin ky) (sqrt (fma (cos (* 2.0 ky)) -0.5 0.5))) (sin th))
     (if (<= t_1 -0.05)
       t_2
       (if (<= t_1 5e-24)
         (* (/ (sin th) (sin kx)) (sin ky))
         (if (<= t_1 0.995) t_2 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_2 = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	double tmp;
	if (t_1 <= -0.996) {
		tmp = (sin(ky) / sqrt(fma(cos((2.0 * ky)), -0.5, 0.5))) * sin(th);
	} else if (t_1 <= -0.05) {
		tmp = t_2;
	} else if (t_1 <= 5e-24) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} else if (t_1 <= 0.995) {
		tmp = t_2;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_2 = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th))
	tmp = 0.0
	if (t_1 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(cos(Float64(2.0 * ky)), -0.5, 0.5))) * sin(th));
	elseif (t_1 <= -0.05)
		tmp = t_2;
	elseif (t_1 <= 5e-24)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	elseif (t_1 <= 0.995)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$2, If[LessEqual[t$95$1, 5e-24], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.995], t$95$2, N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 87.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 61.1

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 59.5

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

   7. Applied rewrites 59.5%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003 or 4.9999999999999998e-24 < (/.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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 52.1

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

   9. Applied rewrites 52.1%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-24

   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

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

      7. lower-/.f64 99.2

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 63.6

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

   7. Applied rewrites 63.6%

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

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

   1. Initial program 87.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot ky\right), -0.5, 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.246:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot ky\right), -0.5, 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.17:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.246)
     (* (/ (sin ky) (sqrt (fma (cos (* 2.0 ky)) -0.5 0.5))) (sin th))
     (if (<= t_1 0.17) (/ (sin th) (/ (sin kx) (sin ky))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.246) {
		tmp = (sin(ky) / sqrt(fma(cos((2.0 * ky)), -0.5, 0.5))) * sin(th);
	} else if (t_1 <= 0.17) {
		tmp = sin(th) / (sin(kx) / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.246)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(cos(Float64(2.0 * ky)), -0.5, 0.5))) * sin(th));
	elseif (t_1 <= 0.17)
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.246], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.17], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.246

   1. Initial program 91.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.8

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 46.6

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

   7. Applied rewrites 46.6%

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

   if -0.246 < (/.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))))) < 0.170000000000000012

   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if 0.170000000000000012 < (/.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 71.9

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

   5. Applied rewrites 71.9%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.246:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot ky\right), -0.5, 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.17:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.17:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.17)
   (/ (sin th) (/ (sin kx) (sin ky)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 0.17) {
		tmp = sin(th) / (sin(kx) / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 0.17d0) then
        tmp = sin(th) / (sin(kx) / sin(ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 0.17) {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 0.17:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.17)
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.17)
		tmp = sin(th) / (sin(kx) / sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.17], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 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))))) < 0.170000000000000012

   1. Initial program 95.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 35.1

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

   7. Applied rewrites 35.1%

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

   if 0.170000000000000012 < (/.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 71.9

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

   5. Applied rewrites 71.9%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.17:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.17:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.17)
   (* (/ (sin th) (sin kx)) (sin ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 0.17) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 0.17d0) then
        tmp = (sin(th) / sin(kx)) * sin(ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 0.17) {
		tmp = (Math.sin(th) / Math.sin(kx)) * Math.sin(ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 0.17:
		tmp = (math.sin(th) / math.sin(kx)) * math.sin(ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.17)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.17)
		tmp = (sin(th) / sin(kx)) * sin(ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.17], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.170000000000000012

   1. Initial program 95.2%

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

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

      7. lower-/.f64 95.2

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 35.1

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

   7. Applied rewrites 35.1%

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

   if 0.170000000000000012 < (/.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 71.9

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

   5. Applied rewrites 71.9%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.17:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.17:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.17)
   (* (/ (sin ky) (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 0.17) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 0.17d0) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 0.17) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 0.17:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.17)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.17)
		tmp = (sin(ky) / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.17], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 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))))) < 0.170000000000000012

   1. Initial program 95.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 35.1

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

   5. Applied rewrites 35.1%

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

   if 0.170000000000000012 < (/.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 71.9

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

   5. Applied rewrites 71.9%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.17:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 4e-7)
   (/ (sin th) (/ (sin kx) ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 4e-7) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 4d-7) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 4e-7) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 4e-7:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 4e-7)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 4e-7)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-7], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-7

   1. Initial program 95.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 33.4

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

   7. Applied rewrites 33.4%

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

   if 3.9999999999999998e-7 < (/.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 91.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 4e-7)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 4e-7) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 4d-7) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 4e-7) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 4e-7:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 4e-7)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 4e-7)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-7], 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))))) < 3.9999999999999998e-7

   1. Initial program 95.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 33.4

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

   5. Applied rewrites 33.4%

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

   if 3.9999999999999998e-7 < (/.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 91.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 31.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 4 \cdot 10^{-74}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 4e-74)
   (* (pow th 3.0) -0.16666666666666666)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 4e-74) {
		tmp = pow(th, 3.0) * -0.16666666666666666;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 4d-74) then
        tmp = (th ** 3.0d0) * (-0.16666666666666666d0)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 4e-74) {
		tmp = Math.pow(th, 3.0) * -0.16666666666666666;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 4e-74:
		tmp = math.pow(th, 3.0) * -0.16666666666666666
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 4e-74)
		tmp = Float64((th ^ 3.0) * -0.16666666666666666);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 4e-74)
		tmp = (th ^ 3.0) * -0.16666666666666666;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-74], N[(N[Power[th, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.99999999999999983e-74

   1. Initial program 94.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 3.8

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

   5. Applied rewrites 3.8%

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

   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 4.0%

      \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 17.0%

       \[\leadsto {th}^{3} \cdot -0.16666666666666666 \]

   if 3.99999999999999983e-74 < (/.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 92.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 60.7

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

   5. Applied rewrites 60.7%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 4 \cdot 10^{-74}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.6 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, \sin kx, \frac{0.5}{\sin kx}\right) \cdot ky, ky, \sin kx\right)}{ky}}\\ \mathbf{elif}\;ky \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + {\sin kx}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right) + \left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.6e-224)
   (/
    (sin th)
    (/
     (fma
      (* (fma 0.16666666666666666 (sin kx) (/ 0.5 (sin kx))) ky)
      ky
      (sin kx))
     ky))
   (if (<= ky 6.6e-5)
     (* (/ (sin ky) (sqrt (+ (* ky ky) (pow (sin kx) 2.0)))) (sin th))
     (*
      (/
       (sin ky)
       (sqrt
        (+ (- 0.5 (* (cos (* 2.0 ky)) 0.5)) (- 0.5 (* (cos (* 2.0 kx)) 0.5)))))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.6e-224) {
		tmp = sin(th) / (fma((fma(0.16666666666666666, sin(kx), (0.5 / sin(kx))) * ky), ky, sin(kx)) / ky);
	} else if (ky <= 6.6e-5) {
		tmp = (sin(ky) / sqrt(((ky * ky) + pow(sin(kx), 2.0)))) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * ky)) * 0.5)) + (0.5 - (cos((2.0 * kx)) * 0.5))))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.6e-224)
		tmp = Float64(sin(th) / Float64(fma(Float64(fma(0.16666666666666666, sin(kx), Float64(0.5 / sin(kx))) * ky), ky, sin(kx)) / ky));
	elseif (ky <= 6.6e-5)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + (sin(kx) ^ 2.0)))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5)) + Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5))))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.6e-224], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[(N[(0.16666666666666666 * N[Sin[kx], $MachinePrecision] + N[(0.5 / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] * ky + N[Sin[kx], $MachinePrecision]), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 6.6e-5], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 1.5999999999999999e-224

   1. Initial program 90.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 90.6

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 30.0%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, \sin kx, \frac{0.5}{\sin kx}\right) \cdot ky, ky, \sin kx\right)}{ky}}} \]

   if 1.5999999999999999e-224 < ky < 6.6000000000000005e-5

   1. Initial program 96.7%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if 6.6000000000000005e-5 < ky

   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

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. lower-*.f64 99.0

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

   6. Applied rewrites 99.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + \left(0.5 - \cos \left(ky \cdot 2\right) \cdot 0.5\right)}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 1.6 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, \sin kx, \frac{0.5}{\sin kx}\right) \cdot ky, ky, \sin kx\right)}{ky}}\\ \mathbf{elif}\;ky \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + {\sin kx}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right) + \left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.35 \cdot 10^{-177}:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{ky}{kx} \cdot 0.5, ky, \sin kx\right)} \cdot \sin ky\\ \mathbf{elif}\;ky \leq 0.022:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right) + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot ky\right), -0.5, 0.5\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.35e-177)
   (* (/ (sin th) (fma (* (/ ky kx) 0.5) ky (sin kx))) (sin ky))
   (if (<= ky 0.022)
     (*
      (/ (sin ky) (sqrt (+ (- 0.5 (* (cos (* 2.0 kx)) 0.5)) (* ky ky))))
      (sin th))
     (* (/ (sin ky) (sqrt (fma (cos (* 2.0 ky)) -0.5 0.5))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.35e-177) {
		tmp = (sin(th) / fma(((ky / kx) * 0.5), ky, sin(kx))) * sin(ky);
	} else if (ky <= 0.022) {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * kx)) * 0.5)) + (ky * ky)))) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt(fma(cos((2.0 * ky)), -0.5, 0.5))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.35e-177)
		tmp = Float64(Float64(sin(th) / fma(Float64(Float64(ky / kx) * 0.5), ky, sin(kx))) * sin(ky));
	elseif (ky <= 0.022)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5)) + Float64(ky * ky)))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(fma(cos(Float64(2.0 * ky)), -0.5, 0.5))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.35e-177], N[(N[(N[Sin[th], $MachinePrecision] / N[(N[(N[(ky / kx), $MachinePrecision] * 0.5), $MachinePrecision] * ky + N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 0.022], 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[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 1.3500000000000001e-177

   1. Initial program 90.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 90.2

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 31.3

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

   7. Applied rewrites 31.3%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 31.4%

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

   if 1.3500000000000001e-177 < ky < 0.021999999999999999

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. count-2 N/A

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

      9. flip-+ N/A

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

      10. +-inverses N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(\frac{\color{blue}{0}}{kx - kx}\right) \cdot \frac{1}{2}\right) + ky \cdot ky}} \cdot \sin th \]

      11. +-inverses N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-inverses N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(\frac{ky \cdot ky - ky \cdot ky}{\color{blue}{0}}\right) \cdot \frac{1}{2}\right) + ky \cdot ky}} \cdot \sin th \]

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. count-2 N/A

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

      18. lower-*.f64 N/A

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

   7. Applied rewrites 88.1%

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

   if 0.021999999999999999 < ky

   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

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 65.5

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

   7. Applied rewrites 65.5%

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

Alternative 15: 24.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \sin th \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = sin(th)
end function

Java

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

Python

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

Julia

function code(kx, ky, th)
	return sin(th)
end

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[Sin[th], $MachinePrecision]

Derivation

1. Initial program 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 25.0

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

5. Applied rewrites 25.0%

   \[\leadsto \color{blue}{\sin th} \]
6. Add Preprocessing

Alternative 16: 14.2% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (fma (* -0.16666666666666666 (* th th)) th th))

C

double code(double kx, double ky, double th) {
	return fma((-0.16666666666666666 * (th * th)), th, th);
}

Julia

function code(kx, ky, th)
	return fma(Float64(-0.16666666666666666 * Float64(th * th)), th, th)
end

Wolfram

code[kx_, ky_, th_] := N[(N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] * th + th), $MachinePrecision]

Derivation

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 25.0

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

5. Applied rewrites 25.0%

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

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 13.8%

   \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]
9. Step-by-step derivation

10. Applied rewrites 13.8%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(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)))