Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.6% → 99.7%

Time: 9.3s

Alternatives: 20

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 81.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + t\_1}}\\ t_4 := \frac{\sin ky}{\frac{\mathsf{fma}\left(0.16666666666666666, th \cdot th, 1\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.002:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2 + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.98:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 t_1))))
        (t_4
         (/
          (sin ky)
          (/
           (*
            (fma 0.16666666666666666 (* th th) 1.0)
            (hypot (sin kx) (sin ky)))
           th))))
   (if (<= t_3 -0.98)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_3 -0.002)
       t_4
       (if (<= t_3 5e-16)
         (* (/ (sin ky) (sqrt (+ t_2 (* ky ky)))) (sin th))
         (if (<= t_3 0.98) t_4 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + t_1));
	double t_4 = sin(ky) / ((fma(0.16666666666666666, (th * th), 1.0) * hypot(sin(kx), sin(ky))) / th);
	double tmp;
	if (t_3 <= -0.98) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_3 <= -0.002) {
		tmp = t_4;
	} else if (t_3 <= 5e-16) {
		tmp = (sin(ky) / sqrt((t_2 + (ky * ky)))) * sin(th);
	} else if (t_3 <= 0.98) {
		tmp = t_4;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + t_1)))
	t_4 = Float64(sin(ky) / Float64(Float64(fma(0.16666666666666666, Float64(th * th), 1.0) * hypot(sin(kx), sin(ky))) / th))
	tmp = 0.0
	if (t_3 <= -0.98)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_3 <= -0.002)
		tmp = t_4;
	elseif (t_3 <= 5e-16)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(t_2 + Float64(ky * ky)))) * sin(th));
	elseif (t_3 <= 0.98)
		tmp = t_4;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[(0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.002], t$95$4, If[LessEqual[t$95$3, 5e-16], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.98], t$95$4, 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))))) < -0.97999999999999998

   1. Initial program 86.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

   if -0.97999999999999998 < (/.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-3 or 5.0000000000000004e-16 < (/.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))))) < 0.97999999999999998

   1. Initial program 99.4%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 5.5

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

   5. Applied rewrites 5.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 5.5

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

   7. Applied rewrites 5.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 5.5

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

   9. Applied rewrites 5.5%

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

   10. Taylor expanded in th around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-/.f64 N/A

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

   12. Applied rewrites 59.0%

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

   if -2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000004e-16

   1. Initial program 98.7%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 97.7

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

   5. Applied rewrites 97.7%

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

Alternative 3: 81.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_1 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.002:\\ \;\;\;\;{\left(\frac{t\_2}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1 + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.98:\\ \;\;\;\;\frac{\sin ky}{t\_2} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (hypot (sin kx) (sin ky)))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_1 t_3)))))
   (if (<= t_4 -0.98)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th))
     (if (<= t_4 -0.002)
       (pow (/ t_2 (* (sin ky) th)) -1.0)
       (if (<= t_4 5e-16)
         (* (/ (sin ky) (sqrt (+ t_1 (* ky ky)))) (sin th))
         (if (<= t_4 0.98) (* (/ (sin ky) t_2) th) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = hypot(sin(kx), sin(ky));
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_1 + t_3));
	double tmp;
	if (t_4 <= -0.98) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	} else if (t_4 <= -0.002) {
		tmp = pow((t_2 / (sin(ky) * th)), -1.0);
	} else if (t_4 <= 5e-16) {
		tmp = (sin(ky) / sqrt((t_1 + (ky * ky)))) * sin(th);
	} else if (t_4 <= 0.98) {
		tmp = (sin(ky) / t_2) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(kx), 2.0);
	double t_2 = Math.hypot(Math.sin(kx), Math.sin(ky));
	double t_3 = Math.pow(Math.sin(ky), 2.0);
	double t_4 = Math.sin(ky) / Math.sqrt((t_1 + t_3));
	double tmp;
	if (t_4 <= -0.98) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + t_3))) * Math.sin(th);
	} else if (t_4 <= -0.002) {
		tmp = Math.pow((t_2 / (Math.sin(ky) * th)), -1.0);
	} else if (t_4 <= 5e-16) {
		tmp = (Math.sin(ky) / Math.sqrt((t_1 + (ky * ky)))) * Math.sin(th);
	} else if (t_4 <= 0.98) {
		tmp = (Math.sin(ky) / t_2) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(kx), 2.0)
	t_2 = math.hypot(math.sin(kx), math.sin(ky))
	t_3 = math.pow(math.sin(ky), 2.0)
	t_4 = math.sin(ky) / math.sqrt((t_1 + t_3))
	tmp = 0
	if t_4 <= -0.98:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + t_3))) * math.sin(th)
	elif t_4 <= -0.002:
		tmp = math.pow((t_2 / (math.sin(ky) * th)), -1.0)
	elif t_4 <= 5e-16:
		tmp = (math.sin(ky) / math.sqrt((t_1 + (ky * ky)))) * math.sin(th)
	elif t_4 <= 0.98:
		tmp = (math.sin(ky) / t_2) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = hypot(sin(kx), sin(ky))
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_1 + t_3)))
	tmp = 0.0
	if (t_4 <= -0.98)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th));
	elseif (t_4 <= -0.002)
		tmp = Float64(t_2 / Float64(sin(ky) * th)) ^ -1.0;
	elseif (t_4 <= 5e-16)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(t_1 + Float64(ky * ky)))) * sin(th));
	elseif (t_4 <= 0.98)
		tmp = Float64(Float64(sin(ky) / t_2) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0;
	t_2 = hypot(sin(kx), sin(ky));
	t_3 = sin(ky) ^ 2.0;
	t_4 = sin(ky) / sqrt((t_1 + t_3));
	tmp = 0.0;
	if (t_4 <= -0.98)
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	elseif (t_4 <= -0.002)
		tmp = (t_2 / (sin(ky) * th)) ^ -1.0;
	elseif (t_4 <= 5e-16)
		tmp = (sin(ky) / sqrt((t_1 + (ky * ky)))) * sin(th);
	elseif (t_4 <= 0.98)
		tmp = (sin(ky) / t_2) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.002], N[Power[N[(t$95$2 / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$4, 5e-16], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.98], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$2), $MachinePrecision] * th), $MachinePrecision], 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))))) < -0.97999999999999998

   1. Initial program 86.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

   if -0.97999999999999998 < (/.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-3

   1. Initial program 99.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 5.3

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

   5. Applied rewrites 5.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 5.3

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

   7. Applied rewrites 5.3%

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

   8. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 54.5

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

   10. Applied rewrites 54.5%

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

   if -2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000004e-16

   1. Initial program 98.7%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 5.0000000000000004e-16 < (/.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))))) < 0.97999999999999998

   1. Initial program 99.2%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 62.3%

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

   if 0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.002:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.98:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_1 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.002:\\ \;\;\;\;{\left(\frac{t\_2}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin th}{\frac{{t\_1}^{0.5}}{ky}}\\ \mathbf{elif}\;t\_4 \leq 0.98:\\ \;\;\;\;\frac{\sin ky}{t\_2} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (hypot (sin kx) (sin ky)))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_1 t_3)))))
   (if (<= t_4 -0.98)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th))
     (if (<= t_4 -0.002)
       (pow (/ t_2 (* (sin ky) th)) -1.0)
       (if (<= t_4 5e-9)
         (/ (sin th) (/ (pow t_1 0.5) ky))
         (if (<= t_4 0.98) (* (/ (sin ky) t_2) th) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = hypot(sin(kx), sin(ky));
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_1 + t_3));
	double tmp;
	if (t_4 <= -0.98) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	} else if (t_4 <= -0.002) {
		tmp = pow((t_2 / (sin(ky) * th)), -1.0);
	} else if (t_4 <= 5e-9) {
		tmp = sin(th) / (pow(t_1, 0.5) / ky);
	} else if (t_4 <= 0.98) {
		tmp = (sin(ky) / t_2) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(kx), 2.0);
	double t_2 = Math.hypot(Math.sin(kx), Math.sin(ky));
	double t_3 = Math.pow(Math.sin(ky), 2.0);
	double t_4 = Math.sin(ky) / Math.sqrt((t_1 + t_3));
	double tmp;
	if (t_4 <= -0.98) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + t_3))) * Math.sin(th);
	} else if (t_4 <= -0.002) {
		tmp = Math.pow((t_2 / (Math.sin(ky) * th)), -1.0);
	} else if (t_4 <= 5e-9) {
		tmp = Math.sin(th) / (Math.pow(t_1, 0.5) / ky);
	} else if (t_4 <= 0.98) {
		tmp = (Math.sin(ky) / t_2) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(kx), 2.0)
	t_2 = math.hypot(math.sin(kx), math.sin(ky))
	t_3 = math.pow(math.sin(ky), 2.0)
	t_4 = math.sin(ky) / math.sqrt((t_1 + t_3))
	tmp = 0
	if t_4 <= -0.98:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + t_3))) * math.sin(th)
	elif t_4 <= -0.002:
		tmp = math.pow((t_2 / (math.sin(ky) * th)), -1.0)
	elif t_4 <= 5e-9:
		tmp = math.sin(th) / (math.pow(t_1, 0.5) / ky)
	elif t_4 <= 0.98:
		tmp = (math.sin(ky) / t_2) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = hypot(sin(kx), sin(ky))
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_1 + t_3)))
	tmp = 0.0
	if (t_4 <= -0.98)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th));
	elseif (t_4 <= -0.002)
		tmp = Float64(t_2 / Float64(sin(ky) * th)) ^ -1.0;
	elseif (t_4 <= 5e-9)
		tmp = Float64(sin(th) / Float64((t_1 ^ 0.5) / ky));
	elseif (t_4 <= 0.98)
		tmp = Float64(Float64(sin(ky) / t_2) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0;
	t_2 = hypot(sin(kx), sin(ky));
	t_3 = sin(ky) ^ 2.0;
	t_4 = sin(ky) / sqrt((t_1 + t_3));
	tmp = 0.0;
	if (t_4 <= -0.98)
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	elseif (t_4 <= -0.002)
		tmp = (t_2 / (sin(ky) * th)) ^ -1.0;
	elseif (t_4 <= 5e-9)
		tmp = sin(th) / ((t_1 ^ 0.5) / ky);
	elseif (t_4 <= 0.98)
		tmp = (sin(ky) / t_2) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.002], N[Power[N[(t$95$2 / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$4, 5e-9], N[(N[Sin[th], $MachinePrecision] / N[(N[Power[t$95$1, 0.5], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.98], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$2), $MachinePrecision] * th), $MachinePrecision], 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))))) < -0.97999999999999998

   1. Initial program 86.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

   if -0.97999999999999998 < (/.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-3

   1. Initial program 99.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 5.3

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

   5. Applied rewrites 5.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 5.3

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

   7. Applied rewrites 5.3%

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

   8. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 54.5

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

   10. Applied rewrites 54.5%

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

   if -2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 98.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 53.2

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

   7. Applied rewrites 53.2%

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

   9. Applied rewrites 98.7%

      \[\leadsto \frac{\sin th}{\frac{{\left({\sin kx}^{2}\right)}^{0.5}}{ky}} \]

   if 5.0000000000000001e-9 < (/.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))))) < 0.97999999999999998

   1. Initial program 99.2%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 62.3%

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

   if 0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.002:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin th}{\frac{{\left({\sin kx}^{2}\right)}^{0.5}}{ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.98:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\left(\frac{2}{1 - \cos \left(2 \cdot ky\right)}\right)}^{-1}}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.002:\\ \;\;\;\;{\left(\frac{t\_1}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin th}{\frac{{t\_2}^{0.5}}{ky}}\\ \mathbf{elif}\;t\_3 \leq 0.98:\\ \;\;\;\;\frac{\sin ky}{t\_1} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky)))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0))))))
   (if (<= t_3 -0.98)
     (*
      (/
       (sin ky)
       (sqrt (+ (* kx kx) (pow (/ 2.0 (- 1.0 (cos (* 2.0 ky)))) -1.0))))
      (sin th))
     (if (<= t_3 -0.002)
       (pow (/ t_1 (* (sin ky) th)) -1.0)
       (if (<= t_3 5e-9)
         (/ (sin th) (/ (pow t_2 0.5) ky))
         (if (<= t_3 0.98) (* (/ (sin ky) t_1) th) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = (sin(ky) / sqrt(((kx * kx) + pow((2.0 / (1.0 - cos((2.0 * ky)))), -1.0)))) * sin(th);
	} else if (t_3 <= -0.002) {
		tmp = pow((t_1 / (sin(ky) * th)), -1.0);
	} else if (t_3 <= 5e-9) {
		tmp = sin(th) / (pow(t_2, 0.5) / ky);
	} else if (t_3 <= 0.98) {
		tmp = (sin(ky) / t_1) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(kx), Math.sin(ky));
	double t_2 = Math.pow(Math.sin(kx), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((t_2 + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + Math.pow((2.0 / (1.0 - Math.cos((2.0 * ky)))), -1.0)))) * Math.sin(th);
	} else if (t_3 <= -0.002) {
		tmp = Math.pow((t_1 / (Math.sin(ky) * th)), -1.0);
	} else if (t_3 <= 5e-9) {
		tmp = Math.sin(th) / (Math.pow(t_2, 0.5) / ky);
	} else if (t_3 <= 0.98) {
		tmp = (Math.sin(ky) / t_1) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(kx), math.sin(ky))
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = math.sin(ky) / math.sqrt((t_2 + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_3 <= -0.98:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + math.pow((2.0 / (1.0 - math.cos((2.0 * ky)))), -1.0)))) * math.sin(th)
	elif t_3 <= -0.002:
		tmp = math.pow((t_1 / (math.sin(ky) * th)), -1.0)
	elif t_3 <= 5e-9:
		tmp = math.sin(th) / (math.pow(t_2, 0.5) / ky)
	elif t_3 <= 0.98:
		tmp = (math.sin(ky) / t_1) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.98)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + (Float64(2.0 / Float64(1.0 - cos(Float64(2.0 * ky)))) ^ -1.0)))) * sin(th));
	elseif (t_3 <= -0.002)
		tmp = Float64(t_1 / Float64(sin(ky) * th)) ^ -1.0;
	elseif (t_3 <= 5e-9)
		tmp = Float64(sin(th) / Float64((t_2 ^ 0.5) / ky));
	elseif (t_3 <= 0.98)
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky));
	t_2 = sin(kx) ^ 2.0;
	t_3 = sin(ky) / sqrt((t_2 + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -0.98)
		tmp = (sin(ky) / sqrt(((kx * kx) + ((2.0 / (1.0 - cos((2.0 * ky)))) ^ -1.0)))) * sin(th);
	elseif (t_3 <= -0.002)
		tmp = (t_1 / (sin(ky) * th)) ^ -1.0;
	elseif (t_3 <= 5e-9)
		tmp = sin(th) / ((t_2 ^ 0.5) / ky);
	elseif (t_3 <= 0.98)
		tmp = (sin(ky) / t_1) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[Power[N[(2.0 / N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.002], N[Power[N[(t$95$1 / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$3, 5e-9], N[(N[Sin[th], $MachinePrecision] / N[(N[Power[t$95$2, 0.5], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.98], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision], 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))))) < -0.97999999999999998

   1. Initial program 86.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. cos-diff N/A

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

      10. cos-sin-sum N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   if -0.97999999999999998 < (/.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-3

   1. Initial program 99.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 5.3

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

   5. Applied rewrites 5.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 5.3

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

   7. Applied rewrites 5.3%

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

   8. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 54.5

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

   10. Applied rewrites 54.5%

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

   if -2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 98.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 53.2

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

   7. Applied rewrites 53.2%

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

   9. Applied rewrites 98.7%

      \[\leadsto \frac{\sin th}{\frac{{\left({\sin kx}^{2}\right)}^{0.5}}{ky}} \]

   if 5.0000000000000001e-9 < (/.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))))) < 0.97999999999999998

   1. Initial program 99.2%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 62.3%

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

   if 0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\left(\frac{2}{1 - \cos \left(2 \cdot ky\right)}\right)}^{-1}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.002:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin th}{\frac{{\left({\sin kx}^{2}\right)}^{0.5}}{ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.98:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\left(\frac{2}{1 - \cos \left(2 \cdot ky\right)}\right)}^{-1}}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin th}{\frac{{t\_2}^{0.5}}{ky}}\\ \mathbf{elif}\;t\_3 \leq 0.98:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin kx) (sin ky))) th))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0))))))
   (if (<= t_3 -0.98)
     (*
      (/
       (sin ky)
       (sqrt (+ (* kx kx) (pow (/ 2.0 (- 1.0 (cos (* 2.0 ky)))) -1.0))))
      (sin th))
     (if (<= t_3 -0.002)
       t_1
       (if (<= t_3 5e-9)
         (/ (sin th) (/ (pow t_2 0.5) ky))
         (if (<= t_3 0.98) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = (sin(ky) / sqrt(((kx * kx) + pow((2.0 / (1.0 - cos((2.0 * ky)))), -1.0)))) * sin(th);
	} else if (t_3 <= -0.002) {
		tmp = t_1;
	} else if (t_3 <= 5e-9) {
		tmp = sin(th) / (pow(t_2, 0.5) / ky);
	} else if (t_3 <= 0.98) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky))) * th;
	double t_2 = Math.pow(Math.sin(kx), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((t_2 + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + Math.pow((2.0 / (1.0 - Math.cos((2.0 * ky)))), -1.0)))) * Math.sin(th);
	} else if (t_3 <= -0.002) {
		tmp = t_1;
	} else if (t_3 <= 5e-9) {
		tmp = Math.sin(th) / (Math.pow(t_2, 0.5) / ky);
	} else if (t_3 <= 0.98) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky))) * th
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = math.sin(ky) / math.sqrt((t_2 + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_3 <= -0.98:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + math.pow((2.0 / (1.0 - math.cos((2.0 * ky)))), -1.0)))) * math.sin(th)
	elif t_3 <= -0.002:
		tmp = t_1
	elif t_3 <= 5e-9:
		tmp = math.sin(th) / (math.pow(t_2, 0.5) / ky)
	elif t_3 <= 0.98:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * th)
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.98)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + (Float64(2.0 / Float64(1.0 - cos(Float64(2.0 * ky)))) ^ -1.0)))) * sin(th));
	elseif (t_3 <= -0.002)
		tmp = t_1;
	elseif (t_3 <= 5e-9)
		tmp = Float64(sin(th) / Float64((t_2 ^ 0.5) / ky));
	elseif (t_3 <= 0.98)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	t_2 = sin(kx) ^ 2.0;
	t_3 = sin(ky) / sqrt((t_2 + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -0.98)
		tmp = (sin(ky) / sqrt(((kx * kx) + ((2.0 / (1.0 - cos((2.0 * ky)))) ^ -1.0)))) * sin(th);
	elseif (t_3 <= -0.002)
		tmp = t_1;
	elseif (t_3 <= 5e-9)
		tmp = sin(th) / ((t_2 ^ 0.5) / ky);
	elseif (t_3 <= 0.98)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[Power[N[(2.0 / N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.002], t$95$1, If[LessEqual[t$95$3, 5e-9], N[(N[Sin[th], $MachinePrecision] / N[(N[Power[t$95$2, 0.5], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.98], t$95$1, 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))))) < -0.97999999999999998

   1. Initial program 86.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. cos-diff N/A

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

      10. cos-sin-sum N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   if -0.97999999999999998 < (/.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-3 or 5.0000000000000001e-9 < (/.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))))) < 0.97999999999999998

   1. Initial program 99.4%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 57.6

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 57.6%

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

   if -2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 98.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 53.2

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

   7. Applied rewrites 53.2%

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

   9. Applied rewrites 98.7%

      \[\leadsto \frac{\sin th}{\frac{{\left({\sin kx}^{2}\right)}^{0.5}}{ky}} \]

   if 0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\left(\frac{2}{1 - \cos \left(2 \cdot ky\right)}\right)}^{-1}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.002:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin th}{\frac{{\left({\sin kx}^{2}\right)}^{0.5}}{ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.98:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\left(\frac{2}{1 - \cos \left(2 \cdot ky\right)}\right)}^{-1}}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;t\_2 \leq 0.98:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin kx) (sin ky))) th))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.98)
     (*
      (/
       (sin ky)
       (sqrt (+ (* kx kx) (pow (/ 2.0 (- 1.0 (cos (* 2.0 ky)))) -1.0))))
      (sin th))
     (if (<= t_2 -0.002)
       t_1
       (if (<= t_2 5e-9)
         (/ (* (sin th) ky) (hypot (sin ky) (sin kx)))
         (if (<= t_2 0.98) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.98) {
		tmp = (sin(ky) / sqrt(((kx * kx) + pow((2.0 / (1.0 - cos((2.0 * ky)))), -1.0)))) * sin(th);
	} else if (t_2 <= -0.002) {
		tmp = t_1;
	} else if (t_2 <= 5e-9) {
		tmp = (sin(th) * ky) / hypot(sin(ky), sin(kx));
	} else if (t_2 <= 0.98) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky))) * th;
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.98) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + Math.pow((2.0 / (1.0 - Math.cos((2.0 * ky)))), -1.0)))) * Math.sin(th);
	} else if (t_2 <= -0.002) {
		tmp = t_1;
	} else if (t_2 <= 5e-9) {
		tmp = (Math.sin(th) * ky) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else if (t_2 <= 0.98) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky))) * th
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= -0.98:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + math.pow((2.0 / (1.0 - math.cos((2.0 * ky)))), -1.0)))) * math.sin(th)
	elif t_2 <= -0.002:
		tmp = t_1
	elif t_2 <= 5e-9:
		tmp = (math.sin(th) * ky) / math.hypot(math.sin(ky), math.sin(kx))
	elif t_2 <= 0.98:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * th)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.98)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + (Float64(2.0 / Float64(1.0 - cos(Float64(2.0 * ky)))) ^ -1.0)))) * sin(th));
	elseif (t_2 <= -0.002)
		tmp = t_1;
	elseif (t_2 <= 5e-9)
		tmp = Float64(Float64(sin(th) * ky) / hypot(sin(ky), sin(kx)));
	elseif (t_2 <= 0.98)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.98)
		tmp = (sin(ky) / sqrt(((kx * kx) + ((2.0 / (1.0 - cos((2.0 * ky)))) ^ -1.0)))) * sin(th);
	elseif (t_2 <= -0.002)
		tmp = t_1;
	elseif (t_2 <= 5e-9)
		tmp = (sin(th) * ky) / hypot(sin(ky), sin(kx));
	elseif (t_2 <= 0.98)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[Power[N[(2.0 / N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.002], t$95$1, If[LessEqual[t$95$2, 5e-9], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.98], t$95$1, 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))))) < -0.97999999999999998

   1. Initial program 86.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. cos-diff N/A

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

      10. cos-sin-sum N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   if -0.97999999999999998 < (/.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-3 or 5.0000000000000001e-9 < (/.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))))) < 0.97999999999999998

   1. Initial program 99.4%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 57.6

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 57.6%

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

   if -2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.6

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 94.8

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

   4. Applied rewrites 94.8%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 94.8

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

   7. Applied rewrites 94.8%

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

   if 0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\left(\frac{2}{1 - \cos \left(2 \cdot ky\right)}\right)}^{-1}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.002:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.98:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\left(\frac{2}{1 - \cos \left(2 \cdot ky\right)}\right)}^{-1}}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.98:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin kx) (sin ky))) th))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.98)
     (*
      (/
       (sin ky)
       (sqrt (+ (* kx kx) (pow (/ 2.0 (- 1.0 (cos (* 2.0 ky)))) -1.0))))
      (sin th))
     (if (<= t_2 -0.002)
       t_1
       (if (<= t_2 5e-9)
         (* (sin ky) (/ (sin th) (sin kx)))
         (if (<= t_2 0.98) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.98) {
		tmp = (sin(ky) / sqrt(((kx * kx) + pow((2.0 / (1.0 - cos((2.0 * ky)))), -1.0)))) * sin(th);
	} else if (t_2 <= -0.002) {
		tmp = t_1;
	} else if (t_2 <= 5e-9) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (t_2 <= 0.98) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky))) * th;
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.98) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + Math.pow((2.0 / (1.0 - Math.cos((2.0 * ky)))), -1.0)))) * Math.sin(th);
	} else if (t_2 <= -0.002) {
		tmp = t_1;
	} else if (t_2 <= 5e-9) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (t_2 <= 0.98) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky))) * th
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= -0.98:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + math.pow((2.0 / (1.0 - math.cos((2.0 * ky)))), -1.0)))) * math.sin(th)
	elif t_2 <= -0.002:
		tmp = t_1
	elif t_2 <= 5e-9:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif t_2 <= 0.98:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * th)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.98)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + (Float64(2.0 / Float64(1.0 - cos(Float64(2.0 * ky)))) ^ -1.0)))) * sin(th));
	elseif (t_2 <= -0.002)
		tmp = t_1;
	elseif (t_2 <= 5e-9)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (t_2 <= 0.98)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.98)
		tmp = (sin(ky) / sqrt(((kx * kx) + ((2.0 / (1.0 - cos((2.0 * ky)))) ^ -1.0)))) * sin(th);
	elseif (t_2 <= -0.002)
		tmp = t_1;
	elseif (t_2 <= 5e-9)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (t_2 <= 0.98)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[Power[N[(2.0 / N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.002], t$95$1, If[LessEqual[t$95$2, 5e-9], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.98], t$95$1, 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))))) < -0.97999999999999998

   1. Initial program 86.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. cos-diff N/A

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

      10. cos-sin-sum N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   if -0.97999999999999998 < (/.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-3 or 5.0000000000000001e-9 < (/.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))))) < 0.97999999999999998

   1. Initial program 99.4%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 57.6

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 57.6%

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

   if -2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 98.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-sin.f64 53.2

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

   5. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 53.3

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

   7. Applied rewrites 53.3%

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

   if 0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\left(\frac{2}{1 - \cos \left(2 \cdot ky\right)}\right)}^{-1}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.002:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.98:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\left(\frac{2}{1 - \cos \left(2 \cdot ky\right)}\right)}^{-1}}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.98:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ th (hypot (sin kx) (sin ky))) (sin ky)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.98)
     (*
      (/
       (sin ky)
       (sqrt (+ (* kx kx) (pow (/ 2.0 (- 1.0 (cos (* 2.0 ky)))) -1.0))))
      (sin th))
     (if (<= t_2 -0.002)
       t_1
       (if (<= t_2 5e-16)
         (* (sin ky) (/ (sin th) (sin kx)))
         (if (<= t_2 0.98) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.98) {
		tmp = (sin(ky) / sqrt(((kx * kx) + pow((2.0 / (1.0 - cos((2.0 * ky)))), -1.0)))) * sin(th);
	} else if (t_2 <= -0.002) {
		tmp = t_1;
	} else if (t_2 <= 5e-16) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (t_2 <= 0.98) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (th / Math.hypot(Math.sin(kx), Math.sin(ky))) * Math.sin(ky);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.98) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + Math.pow((2.0 / (1.0 - Math.cos((2.0 * ky)))), -1.0)))) * Math.sin(th);
	} else if (t_2 <= -0.002) {
		tmp = t_1;
	} else if (t_2 <= 5e-16) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (t_2 <= 0.98) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (th / math.hypot(math.sin(kx), math.sin(ky))) * math.sin(ky)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= -0.98:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + math.pow((2.0 / (1.0 - math.cos((2.0 * ky)))), -1.0)))) * math.sin(th)
	elif t_2 <= -0.002:
		tmp = t_1
	elif t_2 <= 5e-16:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif t_2 <= 0.98:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(th / hypot(sin(kx), sin(ky))) * sin(ky))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.98)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + (Float64(2.0 / Float64(1.0 - cos(Float64(2.0 * ky)))) ^ -1.0)))) * sin(th));
	elseif (t_2 <= -0.002)
		tmp = t_1;
	elseif (t_2 <= 5e-16)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (t_2 <= 0.98)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.98)
		tmp = (sin(ky) / sqrt(((kx * kx) + ((2.0 / (1.0 - cos((2.0 * ky)))) ^ -1.0)))) * sin(th);
	elseif (t_2 <= -0.002)
		tmp = t_1;
	elseif (t_2 <= 5e-16)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (t_2 <= 0.98)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[Power[N[(2.0 / N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.002], t$95$1, If[LessEqual[t$95$2, 5e-16], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.98], t$95$1, 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))))) < -0.97999999999999998

   1. Initial program 86.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. cos-diff N/A

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

      10. cos-sin-sum N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   if -0.97999999999999998 < (/.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-3 or 5.0000000000000004e-16 < (/.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))))) < 0.97999999999999998

   1. Initial program 99.4%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 57.6

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 57.5%

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

   if -2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000004e-16

   1. Initial program 98.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-sin.f64 53.2

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

   5. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 53.3

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

   7. Applied rewrites 53.3%

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

   if 0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\left(\frac{2}{1 - \cos \left(2 \cdot ky\right)}\right)}^{-1}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.002:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.98:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\left(\frac{2}{1 - \cos \left(2 \cdot ky\right)}\right)}^{-1}}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -0.99)
     (*
      (/
       (sin ky)
       (sqrt (+ (* kx kx) (pow (/ 2.0 (- 1.0 (cos (* 2.0 ky)))) -1.0))))
      (sin th))
     (if (<= t_1 5e-9) (* (sin ky) (/ (sin th) (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 <= -0.99) {
		tmp = (sin(ky) / sqrt(((kx * kx) + pow((2.0 / (1.0 - cos((2.0 * ky)))), -1.0)))) * sin(th);
	} else if (t_1 <= 5e-9) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= (-0.99d0)) then
        tmp = (sin(ky) / sqrt(((kx * kx) + ((2.0d0 / (1.0d0 - cos((2.0d0 * ky)))) ** (-1.0d0))))) * sin(th)
    else if (t_1 <= 5d-9) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.99) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + Math.pow((2.0 / (1.0 - Math.cos((2.0 * ky)))), -1.0)))) * Math.sin(th);
	} else if (t_1 <= 5e-9) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.99:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + math.pow((2.0 / (1.0 - math.cos((2.0 * ky)))), -1.0)))) * math.sin(th)
	elif t_1 <= 5e-9:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.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 <= -0.99)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + (Float64(2.0 / Float64(1.0 - cos(Float64(2.0 * ky)))) ^ -1.0)))) * sin(th));
	elseif (t_1 <= 5e-9)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.99)
		tmp = (sin(ky) / sqrt(((kx * kx) + ((2.0 / (1.0 - cos((2.0 * ky)))) ^ -1.0)))) * sin(th);
	elseif (t_1 <= 5e-9)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = 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, -0.99], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[Power[N[(2.0 / N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-9], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 86.0%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. cos-diff N/A

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

      10. cos-sin-sum N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 56.3

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

   7. Applied rewrites 56.3%

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

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 98.9%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 41.3

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

   5. Applied rewrites 41.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.3

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

   7. Applied rewrites 41.3%

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

   if 5.0000000000000001e-9 < (/.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 94.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 77.0

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

   5. Applied rewrites 77.0%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\left(\frac{2}{1 - \cos \left(2 \cdot ky\right)}\right)}^{-1}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -0.99)
     (*
      (/ (sin ky) (sqrt (+ (* kx kx) (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
      (sin th))
     (if (<= t_1 5e-9) (* (sin ky) (/ (sin th) (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 <= -0.99) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * sin(th);
	} else if (t_1 <= 5e-9) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= (-0.99d0)) then
        tmp = (sin(ky) / sqrt(((kx * kx) + (0.5d0 - (cos((2.0d0 * ky)) * 0.5d0))))) * sin(th)
    else if (t_1 <= 5d-9) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.99) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + (0.5 - (Math.cos((2.0 * ky)) * 0.5))))) * Math.sin(th);
	} else if (t_1 <= 5e-9) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.99:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + (0.5 - (math.cos((2.0 * ky)) * 0.5))))) * math.sin(th)
	elif t_1 <= 5e-9:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.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 <= -0.99)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * sin(th));
	elseif (t_1 <= 5e-9)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.99)
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * sin(th);
	elseif (t_1 <= 5e-9)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = 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, -0.99], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-9], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 86.0%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. count-2 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. lower-*.f64 56.2

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

   7. Applied rewrites 56.2%

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

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 98.9%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 41.3

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

   5. Applied rewrites 41.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.3

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

   7. Applied rewrites 41.3%

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

   if 5.0000000000000001e-9 < (/.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 94.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 77.0

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

   5. Applied rewrites 77.0%

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

Alternative 12: 45.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-9)
   (* (sin ky) (/ (sin th) (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5e-9) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 5d-9) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 5e-9) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 5e-9:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 95.2%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 31.7

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

   5. Applied rewrites 31.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 31.7

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

   7. Applied rewrites 31.7%

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

   if 5.0000000000000001e-9 < (/.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 94.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 77.0

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

   5. Applied rewrites 77.0%

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

Alternative 13: 44.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-9)
   (/ (sin th) (/ (sin kx) ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5e-9) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 5d-9) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 5e-9) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 5e-9:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.2

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 29.5

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

   7. Applied rewrites 29.5%

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

   if 5.0000000000000001e-9 < (/.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 94.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 77.0

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

   5. Applied rewrites 77.0%

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

Alternative 14: 44.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-9)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5e-9) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 5d-9) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 5e-9) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 5e-9:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 95.2%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 29.5

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

   5. Applied rewrites 29.5%

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

   if 5.0000000000000001e-9 < (/.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 94.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 77.0

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

   5. Applied rewrites 77.0%

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

Alternative 15: 35.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-23}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1e-23)
   (/ (sin th) (/ kx ky))
   (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)))) <= 1e-23) {
		tmp = sin(th) / (kx / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 1d-23) then
        tmp = sin(th) / (kx / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 1e-23) {
		tmp = Math.sin(th) / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 1e-23:
		tmp = math.sin(th) / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-23)
		tmp = Float64(sin(th) / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-23)
		tmp = sin(th) / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-23], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999996e-24

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.1

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 29.3

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

   7. Applied rewrites 29.3%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 19.5%

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

   if 9.9999999999999996e-24 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.7%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 75.5

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

   5. Applied rewrites 75.5%

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

Alternative 16: 35.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{th}{\sin kx} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-9)
   (* (/ th (sin kx)) ky)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5e-9) {
		tmp = (th / sin(kx)) * ky;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 5d-9) then
        tmp = (th / sin(kx)) * ky
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 5e-9) {
		tmp = (th / Math.sin(kx)) * ky;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 5e-9:
		tmp = (th / math.sin(kx)) * ky
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = Float64(Float64(th / sin(kx)) * ky);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = (th / sin(kx)) * ky;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   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 th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 42.1

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

   5. Applied rewrites 42.1%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 16.5%

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

   if 5.0000000000000001e-9 < (/.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 94.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 77.0

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

   5. Applied rewrites 77.0%

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

Alternative 17: 31.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 3.6 \cdot 10^{-50}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 3.6e-50)
   (* (pow th 3.0) -0.16666666666666666)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 3.6e-50) {
		tmp = pow(th, 3.0) * -0.16666666666666666;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 3.6d-50) then
        tmp = (th ** 3.0d0) * (-0.16666666666666666d0)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 3.6e-50) {
		tmp = Math.pow(th, 3.0) * -0.16666666666666666;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 3.6e-50:
		tmp = math.pow(th, 3.0) * -0.16666666666666666
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 3.6e-50)
		tmp = Float64((th ^ 3.0) * -0.16666666666666666);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 3.6e-50)
		tmp = (th ^ 3.0) * -0.16666666666666666;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.6e-50], N[(N[Power[th, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.59999999999999979e-50

   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. lower-sin.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in th around inf

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

   11. Applied rewrites 12.0%

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

   if 3.59999999999999979e-50 < (/.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 94.7%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 71.1

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

   5. Applied rewrites 71.1%

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

Alternative 18: 71.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.32e-5)
   (/ (* (sin th) ky) (hypot (sin ky) (sin kx)))
   (*
    (/
     (sin ky)
     (/
      (sqrt
       (fma (- 1.0 (cos (* ky 2.0))) 2.0 (* 2.0 (- 1.0 (cos (* 2.0 kx))))))
      2.0))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.32e-5) {
		tmp = (sin(th) * ky) / hypot(sin(ky), sin(kx));
	} else {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((ky * 2.0))), 2.0, (2.0 * (1.0 - cos((2.0 * kx)))))) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.32e-5)
		tmp = Float64(Float64(sin(th) * ky) / hypot(sin(ky), sin(kx)));
	else
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(ky * 2.0))), 2.0, Float64(2.0 * Float64(1.0 - cos(Float64(2.0 * kx)))))) / 2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.32e-5], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(2.0 * N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.32000000000000007e-5

   1. Initial program 93.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.8

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 94.8

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

   4. Applied rewrites 94.8%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 68.4

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

   7. Applied rewrites 68.4%

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

   if 1.32000000000000007e-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. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 23.9% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. lower-sin.f64 27.6

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

5. Applied rewrites 27.6%

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

Alternative 20: 13.5% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (fma (* th th) -0.16666666666666666 1.0) th))

C

double code(double kx, double ky, double th) {
	return fma((th * th), -0.16666666666666666, 1.0) * th;
}

Julia

function code(kx, ky, th)
	return Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th)
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]

Derivation

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. lower-sin.f64 27.6

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

5. Applied rewrites 27.6%

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

6. Taylor expanded in th around 0

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

8. Applied rewrites 16.0%

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

10. Applied rewrites 16.0%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \]
11. Step-by-step derivation

12. Applied rewrites 16.0%

    \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024323 
(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)))