Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.4% → 99.7%

Time: 12.8s

Alternatives: 25

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. accelerator-lowering-hypot.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. sin-lowering-sin.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 82.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.999:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_2 + kx \cdot kx}}\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.86:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (/ (sin ky) (hypot ky (sin kx)))))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -0.999)
     (* (sin th) (/ (sin ky) (sqrt (+ t_2 (* kx kx)))))
     (if (<= t_3 -0.2)
       (*
        (sin ky)
        (/
         th
         (sqrt
          (fma (- 1.0 (cos (+ ky ky))) 0.5 (+ 0.5 (* -0.5 (cos (+ kx kx))))))))
       (if (<= t_3 2e-29)
         t_1
         (if (<= t_3 0.86)
           (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
           (if (<= t_3 2.0) (sin th) t_1)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.999) {
		tmp = sin(th) * (sin(ky) / sqrt((t_2 + (kx * kx))));
	} else if (t_3 <= -0.2) {
		tmp = sin(ky) * (th / sqrt(fma((1.0 - cos((ky + ky))), 0.5, (0.5 + (-0.5 * cos((kx + kx)))))));
	} else if (t_3 <= 2e-29) {
		tmp = t_1;
	} else if (t_3 <= 0.86) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_3 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(ky, sin(kx))))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -0.999)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(t_2 + Float64(kx * kx)))));
	elseif (t_3 <= -0.2)
		tmp = Float64(sin(ky) * Float64(th / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx))))))));
	elseif (t_3 <= 2e-29)
		tmp = t_1;
	elseif (t_3 <= 0.86)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_3 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.999], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-29], t$95$1, If[LessEqual[t$95$3, 0.86], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[Sin[th], $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 82.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 82.5

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

   5. Simplified 82.5%

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

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

   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. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 51.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 51.7%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.99999999999999989e-29 or 2 < (/.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 95.7%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 94.8%

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

   if 1.99999999999999989e-29 < (/.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))))) < 0.859999999999999987

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 65.0%

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

   if 0.859999999999999987 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 87.2

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

   5. Simplified 87.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.999:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin ky}^{2} + kx \cdot kx}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.86:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := 1 - \cos \left(ky + ky\right)\\ t_3 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(t\_2, 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sqrt{\mathsf{fma}\left(t\_2, 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-29}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0.86:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (- 1.0 (cos (+ ky ky))))
        (t_3 (* (sin th) (/ (sin ky) (hypot ky (sin kx))))))
   (if (<= t_1 -0.999)
     (*
      (sin th)
      (/
       (sin ky)
       (sqrt
        (fma t_2 0.5 (* (* kx kx) (fma (* kx kx) -0.3333333333333333 1.0))))))
     (if (<= t_1 -0.2)
       (*
        (sin ky)
        (/ th (sqrt (fma t_2 0.5 (+ 0.5 (* -0.5 (cos (+ kx kx))))))))
       (if (<= t_1 2e-29)
         t_3
         (if (<= t_1 0.86)
           (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
           (if (<= t_1 2.0) (sin th) t_3)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = 1.0 - cos((ky + ky));
	double t_3 = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(t_2, 0.5, ((kx * kx) * fma((kx * kx), -0.3333333333333333, 1.0)))));
	} else if (t_1 <= -0.2) {
		tmp = sin(ky) * (th / sqrt(fma(t_2, 0.5, (0.5 + (-0.5 * cos((kx + kx)))))));
	} else if (t_1 <= 2e-29) {
		tmp = t_3;
	} else if (t_1 <= 0.86) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(1.0 - cos(Float64(ky + ky)))
	t_3 = Float64(sin(th) * Float64(sin(ky) / hypot(ky, sin(kx))))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(t_2, 0.5, Float64(Float64(kx * kx) * fma(Float64(kx * kx), -0.3333333333333333, 1.0))))));
	elseif (t_1 <= -0.2)
		tmp = Float64(sin(ky) * Float64(th / sqrt(fma(t_2, 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx))))))));
	elseif (t_1 <= 2e-29)
		tmp = t_3;
	elseif (t_1 <= 0.86)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_3;
	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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 * 0.5 + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[(t$95$2 * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-29], t$95$3, If[LessEqual[t$95$1, 0.86], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 82.5%

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. count-2 N/A

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

      9. +-inverses N/A

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

      10. cos-0 N/A

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

      11. --lowering--.f64 N/A

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

      12. count-2 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. unpow2 N/A

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

      16. sqr-sin-a N/A

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

      17. cancel-sign-sub-inv N/A

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

      18. +-lowering-+.f64 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. count-2 N/A

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

      22. cos-lowering-cos.f64 N/A

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

      23. +-lowering-+.f64 63.6

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

   4. Applied egg-rr 63.6%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 64.1

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, -0.3333333333333333, 1\right)\right)}} \cdot \sin th \]

   7. Simplified 64.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right)}\right)}} \cdot \sin th \]

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

   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. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 51.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 51.7%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.99999999999999989e-29 or 2 < (/.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 95.7%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 94.8%

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

   if 1.99999999999999989e-29 < (/.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))))) < 0.859999999999999987

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 65.0%

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

   if 0.859999999999999987 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 87.2

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

   5. Simplified 87.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.999:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.86:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := 1 - \cos \left(ky + ky\right)\\ t_3 := \sin ky \cdot \frac{th}{\sqrt{\mathsf{fma}\left(t\_2, 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ t_4 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(t\_2, 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-15}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 0.86:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (- 1.0 (cos (+ ky ky))))
        (t_3
         (*
          (sin ky)
          (/ th (sqrt (fma t_2 0.5 (+ 0.5 (* -0.5 (cos (+ kx kx)))))))))
        (t_4 (* (sin th) (/ (sin ky) (hypot ky (sin kx))))))
   (if (<= t_1 -0.999)
     (*
      (sin th)
      (/
       (sin ky)
       (sqrt
        (fma t_2 0.5 (* (* kx kx) (fma (* kx kx) -0.3333333333333333 1.0))))))
     (if (<= t_1 -0.2)
       t_3
       (if (<= t_1 1e-15)
         t_4
         (if (<= t_1 0.86) t_3 (if (<= t_1 2.0) (sin th) t_4)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = 1.0 - cos((ky + ky));
	double t_3 = sin(ky) * (th / sqrt(fma(t_2, 0.5, (0.5 + (-0.5 * cos((kx + kx)))))));
	double t_4 = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(t_2, 0.5, ((kx * kx) * fma((kx * kx), -0.3333333333333333, 1.0)))));
	} else if (t_1 <= -0.2) {
		tmp = t_3;
	} else if (t_1 <= 1e-15) {
		tmp = t_4;
	} else if (t_1 <= 0.86) {
		tmp = t_3;
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(1.0 - cos(Float64(ky + ky)))
	t_3 = Float64(sin(ky) * Float64(th / sqrt(fma(t_2, 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx))))))))
	t_4 = Float64(sin(th) * Float64(sin(ky) / hypot(ky, sin(kx))))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(t_2, 0.5, Float64(Float64(kx * kx) * fma(Float64(kx * kx), -0.3333333333333333, 1.0))))));
	elseif (t_1 <= -0.2)
		tmp = t_3;
	elseif (t_1 <= 1e-15)
		tmp = t_4;
	elseif (t_1 <= 0.86)
		tmp = t_3;
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_4;
	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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[(t$95$2 * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 * 0.5 + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], t$95$3, If[LessEqual[t$95$1, 1e-15], t$95$4, If[LessEqual[t$95$1, 0.86], t$95$3, If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], t$95$4]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 82.5%

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. count-2 N/A

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

      9. +-inverses N/A

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

      10. cos-0 N/A

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

      11. --lowering--.f64 N/A

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

      12. count-2 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. unpow2 N/A

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

      16. sqr-sin-a N/A

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

      17. cancel-sign-sub-inv N/A

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

      18. +-lowering-+.f64 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. count-2 N/A

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

      22. cos-lowering-cos.f64 N/A

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

      23. +-lowering-+.f64 63.6

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

   4. Applied egg-rr 63.6%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 64.1

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, -0.3333333333333333, 1\right)\right)}} \cdot \sin th \]

   7. Simplified 64.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right)}\right)}} \cdot \sin th \]

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 1.0000000000000001e-15 < (/.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))))) < 0.859999999999999987

   1. Initial program 99.5%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 55.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 55.7%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.0000000000000001e-15 or 2 < (/.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 95.8%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 95.0%

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

   if 0.859999999999999987 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 87.2

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

   5. Simplified 87.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.999:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-15}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.86:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := 1 - \cos \left(ky + ky\right)\\ t_3 := \sin ky \cdot \frac{th}{\sqrt{\mathsf{fma}\left(t\_2, 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(t\_2, 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq -0.5:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 10^{-15}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 0.86:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (- 1.0 (cos (+ ky ky))))
        (t_3
         (*
          (sin ky)
          (/ th (sqrt (fma t_2 0.5 (+ 0.5 (* -0.5 (cos (+ kx kx))))))))))
   (if (<= t_1 -0.999)
     (*
      (sin th)
      (/
       (sin ky)
       (sqrt
        (fma t_2 0.5 (* (* kx kx) (fma (* kx kx) -0.3333333333333333 1.0))))))
     (if (<= t_1 -0.5)
       t_3
       (if (<= t_1 2e-176)
         (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* kx -2.0)) 0.5))))
         (if (<= t_1 1e-15)
           (* (sin th) (/ (sin ky) (sin kx)))
           (if (<= t_1 0.86) t_3 (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = 1.0 - cos((ky + ky));
	double t_3 = sin(ky) * (th / sqrt(fma(t_2, 0.5, (0.5 + (-0.5 * cos((kx + kx)))))));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(t_2, 0.5, ((kx * kx) * fma((kx * kx), -0.3333333333333333, 1.0)))));
	} else if (t_1 <= -0.5) {
		tmp = t_3;
	} else if (t_1 <= 2e-176) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else if (t_1 <= 1e-15) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if (t_1 <= 0.86) {
		tmp = t_3;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(1.0 - cos(Float64(ky + ky)))
	t_3 = Float64(sin(ky) * Float64(th / sqrt(fma(t_2, 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx))))))))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(t_2, 0.5, Float64(Float64(kx * kx) * fma(Float64(kx * kx), -0.3333333333333333, 1.0))))));
	elseif (t_1 <= -0.5)
		tmp = t_3;
	elseif (t_1 <= 2e-176)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	elseif (t_1 <= 1e-15)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif (t_1 <= 0.86)
		tmp = t_3;
	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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[(t$95$2 * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 * 0.5 + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.5], t$95$3, If[LessEqual[t$95$1, 2e-176], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-15], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.86], t$95$3, N[Sin[th], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 82.5%

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. count-2 N/A

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

      9. +-inverses N/A

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

      10. cos-0 N/A

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

      11. --lowering--.f64 N/A

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

      12. count-2 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. unpow2 N/A

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

      16. sqr-sin-a N/A

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

      17. cancel-sign-sub-inv N/A

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

      18. +-lowering-+.f64 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. count-2 N/A

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

      22. cos-lowering-cos.f64 N/A

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

      23. +-lowering-+.f64 63.6

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

   4. Applied egg-rr 63.6%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 64.1

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, -0.3333333333333333, 1\right)\right)}} \cdot \sin th \]

   7. Simplified 64.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right)}\right)}} \cdot \sin th \]

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.5 or 1.0000000000000001e-15 < (/.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))))) < 0.859999999999999987

   1. Initial program 99.5%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 57.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 57.0%

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

   if -0.5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-176

   1. Initial program 99.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 82.5%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 74.6

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

   7. Simplified 74.6%

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

   if 2e-176 < (/.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))))) < 1.0000000000000001e-15

   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

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

      1. sin-lowering-sin.f64 76.8

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

   5. Simplified 76.8%

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

   if 0.859999999999999987 < (/.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 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. sin-lowering-sin.f64 85.2

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

   5. Simplified 85.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.999:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.5:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-15}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.86:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := 1 - \cos \left(ky + ky\right)\\ t_3 := \sin ky \cdot \frac{th}{\sqrt{\mathsf{fma}\left(t\_2, 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(t\_2, 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;t\_1 \leq -0.5:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 10^{-15}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 0.86:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (- 1.0 (cos (+ ky ky))))
        (t_3
         (*
          (sin ky)
          (/ th (sqrt (fma t_2 0.5 (+ 0.5 (* -0.5 (cos (+ kx kx))))))))))
   (if (<= t_1 -0.999)
     (* (sin th) (/ (sin ky) (sqrt (fma t_2 0.5 (* kx kx)))))
     (if (<= t_1 -0.5)
       t_3
       (if (<= t_1 2e-176)
         (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* kx -2.0)) 0.5))))
         (if (<= t_1 1e-15)
           (* (sin th) (/ (sin ky) (sin kx)))
           (if (<= t_1 0.86) t_3 (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = 1.0 - cos((ky + ky));
	double t_3 = sin(ky) * (th / sqrt(fma(t_2, 0.5, (0.5 + (-0.5 * cos((kx + kx)))))));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(t_2, 0.5, (kx * kx))));
	} else if (t_1 <= -0.5) {
		tmp = t_3;
	} else if (t_1 <= 2e-176) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else if (t_1 <= 1e-15) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if (t_1 <= 0.86) {
		tmp = t_3;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(1.0 - cos(Float64(ky + ky)))
	t_3 = Float64(sin(ky) * Float64(th / sqrt(fma(t_2, 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx))))))))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(t_2, 0.5, Float64(kx * kx)))));
	elseif (t_1 <= -0.5)
		tmp = t_3;
	elseif (t_1 <= 2e-176)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	elseif (t_1 <= 1e-15)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif (t_1 <= 0.86)
		tmp = t_3;
	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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[(t$95$2 * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.5], t$95$3, If[LessEqual[t$95$1, 2e-176], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-15], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.86], t$95$3, N[Sin[th], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 82.5%

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. count-2 N/A

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

      9. +-inverses N/A

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

      10. cos-0 N/A

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

      11. --lowering--.f64 N/A

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

      12. count-2 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. unpow2 N/A

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

      16. sqr-sin-a N/A

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

      17. cancel-sign-sub-inv N/A

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

      18. +-lowering-+.f64 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. count-2 N/A

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

      22. cos-lowering-cos.f64 N/A

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

      23. +-lowering-+.f64 63.6

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

   4. Applied egg-rr 63.6%

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

   5. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 64.1

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

   7. Simplified 64.1%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.5 or 1.0000000000000001e-15 < (/.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))))) < 0.859999999999999987

   1. Initial program 99.5%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 57.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 57.0%

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

   if -0.5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-176

   1. Initial program 99.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 82.5%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 74.6

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

   7. Simplified 74.6%

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

   if 2e-176 < (/.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))))) < 1.0000000000000001e-15

   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

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

      1. sin-lowering-sin.f64 76.8

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

   5. Simplified 76.8%

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

   if 0.859999999999999987 < (/.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 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. sin-lowering-sin.f64 85.2

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

   5. Simplified 85.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.999:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.5:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-15}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.86:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.71:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.71)
     (* (sin th) (/ (sin ky) (sqrt (* 0.5 (- 1.0 (cos (* ky -2.0)))))))
     (if (<= t_1 2e-176)
       (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* kx -2.0)) 0.5))))
       (if (<= t_1 0.1) (* (sin th) (/ (sin ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.71) {
		tmp = sin(th) * (sin(ky) / sqrt((0.5 * (1.0 - cos((ky * -2.0))))));
	} else if (t_1 <= 2e-176) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else if (t_1 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.71)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(ky * -2.0)))))));
	elseif (t_1 <= 2e-176)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	elseif (t_1 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.71], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-176], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))))) < -0.70999999999999996

   1. Initial program 86.2%

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. count-2 N/A

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

      9. +-inverses N/A

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

      10. cos-0 N/A

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

      11. --lowering--.f64 N/A

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

      12. count-2 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. unpow2 N/A

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

      16. sqr-sin-a N/A

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

      17. cancel-sign-sub-inv N/A

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

      18. +-lowering-+.f64 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. count-2 N/A

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

      22. cos-lowering-cos.f64 N/A

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

      23. +-lowering-+.f64 71.3

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

   4. Applied egg-rr 71.3%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. --lowering--.f64 N/A

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 54.9

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

   7. Simplified 54.9%

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

   if -0.70999999999999996 < (/.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))))) < 2e-176

   1. Initial program 99.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 83.9%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 70.0

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

   7. Simplified 70.0%

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

   if 2e-176 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 69.1

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

   5. Simplified 69.1%

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

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

   1. Initial program 95.2%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 70.7

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

   5. Simplified 70.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.71:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -1.0)
     (* th (/ (sin ky) (hypot (sin ky) kx)))
     (if (<= t_1 2e-176)
       (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* kx -2.0)) 0.5))))
       (if (<= t_1 0.1) (* (sin th) (/ (sin ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = th * (sin(ky) / hypot(sin(ky), kx));
	} else if (t_1 <= 2e-176) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else if (t_1 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (t_1 <= 2e-176)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	elseif (t_1 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-176], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 82.1%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 43.6%

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

   8. Taylor expanded in kx around 0

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

   10. Simplified 43.6%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-176

   1. Initial program 99.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 85.3%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 61.8

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

   7. Simplified 61.8%

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

   if 2e-176 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 69.1

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

   5. Simplified 69.1%

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

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

   1. Initial program 95.2%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 70.7

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

   5. Simplified 70.7%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -1.0)
     (* th (/ (sin ky) (hypot (sin ky) kx)))
     (if (<= t_1 2e-176)
       (* (sin th) (/ (sin ky) (sqrt (fma -0.5 (cos (* kx -2.0)) 0.5))))
       (if (<= t_1 0.1) (* (sin th) (/ (sin ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = th * (sin(ky) / hypot(sin(ky), kx));
	} else if (t_1 <= 2e-176) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else if (t_1 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (t_1 <= 2e-176)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	elseif (t_1 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-176], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 82.1%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 43.6%

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

   8. Taylor expanded in kx around 0

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

   10. Simplified 43.6%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-176

   1. Initial program 99.7%

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. count-2 N/A

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

      9. +-inverses N/A

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

      10. cos-0 N/A

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

      11. --lowering--.f64 N/A

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

      12. count-2 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. unpow2 N/A

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

      16. sqr-sin-a N/A

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

      17. cancel-sign-sub-inv N/A

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

      18. +-lowering-+.f64 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. count-2 N/A

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

      22. cos-lowering-cos.f64 N/A

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

      23. +-lowering-+.f64 85.2

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

   4. Applied egg-rr 85.2%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 61.9

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

   7. Simplified 61.9%

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

   if 2e-176 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 69.1

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

   5. Simplified 69.1%

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

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

   1. Initial program 95.2%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 70.7

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

   5. Simplified 70.7%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(ky \cdot -2\right)}} \cdot \left(\sin ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.05)
     (*
      th
      (* (sqrt (/ 1.0 (- 1.0 (cos (* ky -2.0))))) (* (sin ky) (sqrt 2.0))))
     (if (<= t_1 2e-176)
       (* (sin th) (* ky (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5)))))
       (if (<= t_1 0.1) (* (sin th) (/ (sin ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = th * (sqrt((1.0 / (1.0 - cos((ky * -2.0))))) * (sin(ky) * sqrt(2.0)));
	} else if (t_1 <= 2e-176) {
		tmp = sin(th) * (ky * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5))));
	} else if (t_1 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.05)
		tmp = Float64(th * Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(ky * -2.0))))) * Float64(sin(ky) * sqrt(2.0))));
	elseif (t_1 <= 2e-176)
		tmp = Float64(sin(th) * Float64(ky * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5)))));
	elseif (t_1 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.05], N[(th * N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-176], N[(N[Sin[th], $MachinePrecision] * N[(ky * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))))) < -0.050000000000000003

   1. Initial program 88.3%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 45.8%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 36.2%

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

   10. Taylor expanded in kx around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sqrt-lowering-sqrt.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. metadata-eval N/A

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

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

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

       7. --lowering--.f64 N/A

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

       8. cos-neg N/A

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

       9. cos-lowering-cos.f64 N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. sin-lowering-sin.f64 N/A

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

       14. sqrt-lowering-sqrt.f64 24.1

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

   12. Simplified 24.1%

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

   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))))) < 2e-176

   1. Initial program 99.8%

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. count-2 N/A

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

      9. +-inverses N/A

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

      10. cos-0 N/A

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

      11. --lowering--.f64 N/A

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

      12. count-2 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. unpow2 N/A

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

      16. sqr-sin-a N/A

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

      17. cancel-sign-sub-inv N/A

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

      18. +-lowering-+.f64 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. count-2 N/A

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

      22. cos-lowering-cos.f64 N/A

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

      23. +-lowering-+.f64 81.3

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. cos-neg N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 76.6

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

   7. Simplified 76.6%

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

   if 2e-176 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 69.1

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

   5. Simplified 69.1%

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

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

   1. Initial program 95.2%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 70.7

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

   5. Simplified 70.7%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(ky \cdot -2\right)}} \cdot \left(\sin ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(ky \cdot -2\right)}} \cdot \left(\sin ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.05)
     (*
      th
      (* (sqrt (/ 1.0 (- 1.0 (cos (* ky -2.0))))) (* (sin ky) (sqrt 2.0))))
     (if (<= t_1 2e-176)
       (* (sin th) (* ky (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5)))))
       (if (<= t_1 0.0001) (* (sin th) (/ ky (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = th * (sqrt((1.0 / (1.0 - cos((ky * -2.0))))) * (sin(ky) * sqrt(2.0)));
	} else if (t_1 <= 2e-176) {
		tmp = sin(th) * (ky * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5))));
	} else if (t_1 <= 0.0001) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.05)
		tmp = Float64(th * Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(ky * -2.0))))) * Float64(sin(ky) * sqrt(2.0))));
	elseif (t_1 <= 2e-176)
		tmp = Float64(sin(th) * Float64(ky * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5)))));
	elseif (t_1 <= 0.0001)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.05], N[(th * N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-176], N[(N[Sin[th], $MachinePrecision] * N[(ky * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))))) < -0.050000000000000003

   1. Initial program 88.3%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 45.8%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 36.2%

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

   10. Taylor expanded in kx around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sqrt-lowering-sqrt.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. metadata-eval N/A

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

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

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

       7. --lowering--.f64 N/A

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

       8. cos-neg N/A

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

       9. cos-lowering-cos.f64 N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. sin-lowering-sin.f64 N/A

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

       14. sqrt-lowering-sqrt.f64 24.1

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

   12. Simplified 24.1%

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

   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))))) < 2e-176

   1. Initial program 99.8%

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. count-2 N/A

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

      9. +-inverses N/A

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

      10. cos-0 N/A

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

      11. --lowering--.f64 N/A

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

      12. count-2 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. unpow2 N/A

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

      16. sqr-sin-a N/A

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

      17. cancel-sign-sub-inv N/A

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

      18. +-lowering-+.f64 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. count-2 N/A

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

      22. cos-lowering-cos.f64 N/A

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

      23. +-lowering-+.f64 81.3

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. cos-neg N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 76.6

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

   7. Simplified 76.6%

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

   if 2e-176 < (/.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))))) < 1.00000000000000005e-4

   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

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 73.7

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

   5. Simplified 73.7%

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

   if 1.00000000000000005e-4 < (/.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 95.3%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 69.4

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

   5. Simplified 69.4%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(ky \cdot -2\right)}} \cdot \left(\sin ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;th \cdot \left(\sin ky \cdot \frac{1}{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.05)
     (* th (* (sin ky) (/ 1.0 (sqrt (* 0.5 (- 1.0 (cos (* ky -2.0))))))))
     (if (<= t_1 2e-176)
       (* (sin th) (* ky (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5)))))
       (if (<= t_1 0.0001) (* (sin th) (/ ky (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = th * (sin(ky) * (1.0 / sqrt((0.5 * (1.0 - cos((ky * -2.0)))))));
	} else if (t_1 <= 2e-176) {
		tmp = sin(th) * (ky * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5))));
	} else if (t_1 <= 0.0001) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.05)
		tmp = Float64(th * Float64(sin(ky) * Float64(1.0 / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(ky * -2.0))))))));
	elseif (t_1 <= 2e-176)
		tmp = Float64(sin(th) * Float64(ky * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5)))));
	elseif (t_1 <= 0.0001)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.05], N[(th * N[(N[Sin[ky], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-176], N[(N[Sin[th], $MachinePrecision] * N[(ky * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))))) < -0.050000000000000003

   1. Initial program 88.3%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 45.8%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 36.2%

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

   10. Taylor expanded in kx around 0

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

       1. *-lowering-*.f64 N/A

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

       2. metadata-eval N/A

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

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

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

       4. --lowering--.f64 N/A

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

       5. cos-neg N/A

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

       6. cos-lowering-cos.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 24.0

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

   12. Simplified 24.0%

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

   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))))) < 2e-176

   1. Initial program 99.8%

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. count-2 N/A

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

      9. +-inverses N/A

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

      10. cos-0 N/A

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

      11. --lowering--.f64 N/A

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

      12. count-2 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. unpow2 N/A

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

      16. sqr-sin-a N/A

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

      17. cancel-sign-sub-inv N/A

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

      18. +-lowering-+.f64 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. count-2 N/A

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

      22. cos-lowering-cos.f64 N/A

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

      23. +-lowering-+.f64 81.3

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. cos-neg N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 76.6

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

   7. Simplified 76.6%

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

   if 2e-176 < (/.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))))) < 1.00000000000000005e-4

   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

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 73.7

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

   5. Simplified 73.7%

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

   if 1.00000000000000005e-4 < (/.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 95.3%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 69.4

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

   5. Simplified 69.4%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;th \cdot \left(\sin ky \cdot \frac{1}{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 2e-176)
     (* (sin th) (* ky (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5)))))
     (if (<= t_1 0.0001) (* (sin th) (/ ky (sin kx))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 2e-176) {
		tmp = sin(th) * (ky * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5))));
	} else if (t_1 <= 0.0001) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 2e-176)
		tmp = Float64(sin(th) * Float64(ky * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5)))));
	elseif (t_1 <= 0.0001)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-176], N[(N[Sin[th], $MachinePrecision] * N[(ky * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $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))))) < 2e-176

   1. Initial program 93.8%

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. count-2 N/A

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

      9. +-inverses N/A

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

      10. cos-0 N/A

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

      11. --lowering--.f64 N/A

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

      12. count-2 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. unpow2 N/A

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

      16. sqr-sin-a N/A

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

      17. cancel-sign-sub-inv N/A

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

      18. +-lowering-+.f64 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. count-2 N/A

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

      22. cos-lowering-cos.f64 N/A

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

      23. +-lowering-+.f64 78.4

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

   4. Applied egg-rr 78.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}}\right) \cdot \sin th \]

      8. cos-neg N/A

         \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}}\right) \cdot \sin th \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}}\right) \cdot \sin th \]

      10. *-commutative N/A

          \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}}\right) \cdot \sin th \]

      11. *-lowering-*.f64 38.4

          \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}}\right) \cdot \sin th \]

   7. Simplified 38.4%

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\right)} \cdot \sin th \]

   if 2e-176 < (/.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))))) < 1.00000000000000005e-4

   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

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. sin-lowering-sin.f64 73.7

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 1.00000000000000005e-4 < (/.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 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 69.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 69.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}} \cdot \left(ky \cdot \sin th\right)\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 2e-176)
     (* (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5))) (* ky (sin th)))
     (if (<= t_1 0.0001) (* (sin th) (/ ky (sin kx))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 2e-176) {
		tmp = sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5))) * (ky * sin(th));
	} else if (t_1 <= 0.0001) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 2e-176)
		tmp = Float64(sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))) * Float64(ky * sin(th)));
	elseif (t_1 <= 0.0001)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-176], N[(N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $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))))) < 2e-176

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. div-inv N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      8. count-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. +-inverses N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \color{blue}{0} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. cos-0 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. --lowering--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      16. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      21. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)}} \cdot \sin th \]

      22. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)}} \cdot \sin th \]

      23. +-lowering-+.f64 78.4

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \color{blue}{\left(kx + kx\right)}\right)}} \cdot \sin th \]

   4. Applied egg-rr 78.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \sin th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \left(ky \cdot \color{blue}{\sin th}\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      6. +-commutative N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2}}}} \]

      7. metadata-eval N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \]

      10. cos-neg N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      12. *-commutative N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \]

      13. *-lowering-*.f64 38.2

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \]

   7. Simplified 38.2%

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \]

   if 2e-176 < (/.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))))) < 1.00000000000000005e-4

   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

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. sin-lowering-sin.f64 73.7

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 1.00000000000000005e-4 < (/.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 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 69.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 69.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}} \cdot \left(ky \cdot \sin th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.0001)
   (* (sin th) (/ ky (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.0001) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.0001d0) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.0001) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.0001:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.0001)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.0001)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000005e-4

   1. Initial program 94.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. sin-lowering-sin.f64 39.2

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 39.2%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 1.00000000000000005e-4 < (/.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 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 69.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 69.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.02)
   (* (sin ky) (/ (sin th) kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.02) {
		tmp = sin(ky) * (sin(th) / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.02d0) then
        tmp = sin(ky) * (sin(th) / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.02) {
		tmp = Math.sin(ky) * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.02:
		tmp = math.sin(ky) * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.02)
		tmp = Float64(sin(ky) * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.02)
		tmp = sin(ky) * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.02], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 94.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin ky \cdot \sin th\right)}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      3. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin ky \cdot \sin th\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin ky \cdot \sin th\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sin th \cdot \sin ky}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right)} \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right)} \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin th} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \left(\sin th \cdot \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right)}\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      11. neg-mul-1 N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{1}{\color{blue}{-1 \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      12. associate-/r* N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \color{blue}{\frac{\frac{1}{-1}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      13. metadata-eval N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{\color{blue}{-1}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \color{blue}{\frac{-1}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      16. +-commutative N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

   4. Applied egg-rr 74.6%

      \[\leadsto \color{blue}{\left(\sin th \cdot \left(-\sin ky\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2}}}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \]

      5. cos-neg N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \]

      8. *-lowering-*.f64 43.3

         \[\leadsto \left(\sin th \cdot \left(-\sin ky\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \]

   7. Simplified 43.3%

      \[\leadsto \left(\sin th \cdot \left(-\sin ky\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{kx}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{kx}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{kx}} \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin ky} \cdot \frac{\sin th}{kx} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{kx}} \]

      5. sin-lowering-sin.f64 24.0

         \[\leadsto \sin ky \cdot \frac{\color{blue}{\sin th}}{kx} \]

   10. Simplified 24.0%

       \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{kx}} \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 69.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 69.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 34.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;\left(\sin th \cdot \left(ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.16666666666666666, -1\right)\right)\right) \cdot \frac{-1}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.0001)
   (* (* (sin th) (* ky (fma (* ky ky) 0.16666666666666666 -1.0))) (/ -1.0 kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.0001) {
		tmp = (sin(th) * (ky * fma((ky * ky), 0.16666666666666666, -1.0))) * (-1.0 / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.0001)
		tmp = Float64(Float64(sin(th) * Float64(ky * fma(Float64(ky * ky), 0.16666666666666666, -1.0))) * Float64(-1.0 / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(N[Sin[th], $MachinePrecision] * N[(ky * N[(N[(ky * ky), $MachinePrecision] * 0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000005e-4

   1. Initial program 94.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin ky \cdot \sin th\right)}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      3. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin ky \cdot \sin th\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin ky \cdot \sin th\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sin th \cdot \sin ky}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right)} \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right)} \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin th} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \left(\sin th \cdot \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right)}\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      11. neg-mul-1 N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{1}{\color{blue}{-1 \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      12. associate-/r* N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \color{blue}{\frac{\frac{1}{-1}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      13. metadata-eval N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{\color{blue}{-1}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \color{blue}{\frac{-1}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      16. +-commutative N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

   4. Applied egg-rr 74.6%

      \[\leadsto \color{blue}{\left(\sin th \cdot \left(-\sin ky\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2}}}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \]

      5. cos-neg N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \]

      8. *-lowering-*.f64 43.3

         \[\leadsto \left(\sin th \cdot \left(-\sin ky\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \]

   7. Simplified 43.3%

      \[\leadsto \left(\sin th \cdot \left(-\sin ky\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\color{blue}{kx}} \]
   9. Step-by-step derivation

   10. Simplified 23.5%

       \[\leadsto \left(\sin th \cdot \left(-\sin ky\right)\right) \cdot \frac{-1}{\color{blue}{kx}} \]

   11. Taylor expanded in ky around 0

       \[\leadsto \left(\sin th \cdot \color{blue}{\left(ky \cdot \left(\frac{1}{6} \cdot {ky}^{2} - 1\right)\right)}\right) \cdot \frac{-1}{kx} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \left(\sin th \cdot \color{blue}{\left(ky \cdot \left(\frac{1}{6} \cdot {ky}^{2} - 1\right)\right)}\right) \cdot \frac{-1}{kx} \]

       2. sub-neg N/A

          \[\leadsto \left(\sin th \cdot \left(ky \cdot \color{blue}{\left(\frac{1}{6} \cdot {ky}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \cdot \frac{-1}{kx} \]

       3. *-commutative N/A

          \[\leadsto \left(\sin th \cdot \left(ky \cdot \left(\color{blue}{{ky}^{2} \cdot \frac{1}{6}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \cdot \frac{-1}{kx} \]

       4. metadata-eval N/A

          \[\leadsto \left(\sin th \cdot \left(ky \cdot \left({ky}^{2} \cdot \frac{1}{6} + \color{blue}{-1}\right)\right)\right) \cdot \frac{-1}{kx} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\sin th \cdot \left(ky \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{1}{6}, -1\right)}\right)\right) \cdot \frac{-1}{kx} \]

       6. unpow2 N/A

          \[\leadsto \left(\sin th \cdot \left(ky \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{6}, -1\right)\right)\right) \cdot \frac{-1}{kx} \]

       7. *-lowering-*.f64 23.1

          \[\leadsto \left(\sin th \cdot \left(ky \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, 0.16666666666666666, -1\right)\right)\right) \cdot \frac{-1}{kx} \]

   13. Simplified 23.1%

       \[\leadsto \left(\sin th \cdot \color{blue}{\left(ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.16666666666666666, -1\right)\right)}\right) \cdot \frac{-1}{kx} \]

   if 1.00000000000000005e-4 < (/.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 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 69.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 69.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 15.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-299}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (sin th)
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
      1e-299)
   (* -0.16666666666666666 (* th (* th th)))
   th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(th) * (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))) <= 1e-299) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = 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(th) * (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0))))) <= 1d-299) then
        tmp = (-0.16666666666666666d0) * (th * (th * th))
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(th) * (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0))))) <= 1e-299) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(th) * (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0))))) <= 1e-299:
		tmp = -0.16666666666666666 * (th * (th * th))
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(th) * Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 1e-299)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(th) * (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 1e-299)
		tmp = -0.16666666666666666 * (th * (th * th));
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[th], $MachinePrecision] * 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]), $MachinePrecision], 1e-299], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 9.99999999999999992e-300

   1. Initial program 94.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 22.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 22.9%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. *-lowering-*.f64 14.3

         \[\leadsto \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 14.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   9. Taylor expanded in th around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot \left(th \cdot th\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{{th}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot {th}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

       6. *-lowering-*.f64 17.2

          \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Simplified 17.2%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]

   if 9.99999999999999992e-300 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 30.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 30.7%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th} \]
   7. Step-by-step derivation

   8. Simplified 17.5%

      \[\leadsto \color{blue}{th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-299}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;\frac{ky \cdot \sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.0001)
   (/ (* ky (sin th)) kx)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.0001) {
		tmp = (ky * sin(th)) / kx;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.0001d0) then
        tmp = (ky * sin(th)) / kx
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.0001) {
		tmp = (ky * Math.sin(th)) / kx;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.0001:
		tmp = (ky * math.sin(th)) / kx
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.0001)
		tmp = Float64(Float64(ky * sin(th)) / kx);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.0001)
		tmp = (ky * sin(th)) / kx;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000005e-4

   1. Initial program 94.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin ky \cdot \sin th\right)}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      3. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin ky \cdot \sin th\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin ky \cdot \sin th\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sin th \cdot \sin ky}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right)} \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right)} \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin th} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \left(\sin th \cdot \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right)}\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      11. neg-mul-1 N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{1}{\color{blue}{-1 \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      12. associate-/r* N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \color{blue}{\frac{\frac{1}{-1}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      13. metadata-eval N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{\color{blue}{-1}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \color{blue}{\frac{-1}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      16. +-commutative N/A

          \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

   4. Applied egg-rr 74.6%

      \[\leadsto \color{blue}{\left(\sin th \cdot \left(-\sin ky\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2}}}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \]

      5. cos-neg N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \]

      8. *-lowering-*.f64 43.3

         \[\leadsto \left(\sin th \cdot \left(-\sin ky\right)\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \]

   7. Simplified 43.3%

      \[\leadsto \left(\sin th \cdot \left(-\sin ky\right)\right) \cdot \frac{-1}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\sin th \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)\right) \cdot \frac{-1}{\color{blue}{kx}} \]
   9. Step-by-step derivation

   10. Simplified 23.5%

       \[\leadsto \left(\sin th \cdot \left(-\sin ky\right)\right) \cdot \frac{-1}{\color{blue}{kx}} \]

   11. Taylor expanded in ky around 0

       \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{ky \cdot \sin th}}{kx} \]

       3. sin-lowering-sin.f64 23.1

          \[\leadsto \frac{ky \cdot \color{blue}{\sin th}}{kx} \]

   13. Simplified 23.1%

       \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]

   if 1.00000000000000005e-4 < (/.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 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 69.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 69.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 35.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.0001)
   (* th (/ ky (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.0001) {
		tmp = th * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.0001d0) then
        tmp = th * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.0001) {
		tmp = th * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.0001:
		tmp = th * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.0001)
		tmp = Float64(th * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.0001)
		tmp = th * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0001], N[(th * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000005e-4

   1. Initial program 94.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. sin-lowering-sin.f64 99.8

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   4. Applied egg-rr 99.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Simplified 49.8%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{th} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot th \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot th \]

      2. sin-lowering-sin.f64 22.4

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot th \]

   10. Simplified 22.4%

       \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot th \]

   if 1.00000000000000005e-4 < (/.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 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 69.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 69.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 35.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.0001)
   (* ky (/ th (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.0001) {
		tmp = ky * (th / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.0001d0) then
        tmp = ky * (th / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.0001) {
		tmp = ky * (th / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.0001:
		tmp = ky * (th / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.0001)
		tmp = Float64(ky * Float64(th / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.0001)
		tmp = ky * (th / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0001], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000005e-4

   1. Initial program 94.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. sin-lowering-sin.f64 99.8

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   4. Applied egg-rr 99.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Simplified 49.8%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{th} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot th}{\sin kx}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto ky \cdot \color{blue}{\frac{th}{\sin kx}} \]

      4. sin-lowering-sin.f64 22.3

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

   10. Simplified 22.3%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   if 1.00000000000000005e-4 < (/.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 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 69.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 69.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 30.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-47}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-47)
   (* -0.16666666666666666 (* th (* th th)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-47) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 4d-47) then
        tmp = (-0.16666666666666666d0) * (th * (th * th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 4e-47) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 4e-47:
		tmp = -0.16666666666666666 * (th * (th * th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-47)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-47)
		tmp = -0.16666666666666666 * (th * (th * th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-47], N[(-0.16666666666666666 * N[(th * N[(th * th), $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))))) < 3.9999999999999999e-47

   1. Initial program 94.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 3.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 3.6%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. *-lowering-*.f64 3.6

         \[\leadsto \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 3.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   9. Taylor expanded in th around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot \left(th \cdot th\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{{th}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot {th}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

       6. *-lowering-*.f64 16.2

          \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Simplified 16.2%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]

   if 3.9999999999999999e-47 < (/.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 95.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 66.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 66.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 75.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 5.8e-5)
   (* (sin th) (/ (sin ky) (hypot ky (sin kx))))
   (*
    (sin ky)
    (/
     (sin th)
     (sqrt
      (fma (- 1.0 (cos (+ ky ky))) 0.5 (+ 0.5 (* -0.5 (cos (+ kx kx))))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 5.8e-5) {
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(ky) * (sin(th) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, (0.5 + (-0.5 * cos((kx + kx)))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 5.8e-5)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(ky, sin(kx))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx))))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 5.8e-5], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 5.8e-5

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. sin-lowering-sin.f64 99.8

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   4. Applied egg-rr 99.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Simplified 67.3%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]

   if 5.8e-5 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin ky} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 75.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 5.8e-5)
   (* (sin th) (/ (sin ky) (hypot ky (sin kx))))
   (*
    (sin th)
    (/
     (sin ky)
     (sqrt
      (fma (- 1.0 (cos (+ ky ky))) 0.5 (+ 0.5 (* -0.5 (cos (+ kx kx))))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 5.8e-5) {
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, (0.5 + (-0.5 * cos((kx + kx)))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 5.8e-5)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(ky, sin(kx))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx))))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 5.8e-5], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 5.8e-5

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. sin-lowering-sin.f64 99.8

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   4. Applied egg-rr 99.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Simplified 67.3%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]

   if 5.8e-5 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. div-inv N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      8. count-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. +-inverses N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \color{blue}{0} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. cos-0 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. --lowering--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      16. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      21. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)}} \cdot \sin th \]

      22. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)}} \cdot \sin th \]

      23. +-lowering-+.f64 99.3

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \color{blue}{\left(kx + kx\right)}\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 13.6% accurate, 632.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return th;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 94.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 26.1

      \[\leadsto \color{blue}{\sin th} \]

5. Simplified 26.1%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto \color{blue}{th} \]
7. Step-by-step derivation

8. Simplified 15.5%

   \[\leadsto \color{blue}{th} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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)))