Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 11.9s

Alternatives: 23

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 82.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin kx}^{2}}}\\ t_3 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.04:\\ \;\;\;\;\frac{th}{t\_3} \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{\sin th}{\sqrt{\mathsf{fma}\left(\sin kx, \sin kx, ky \cdot ky\right)}} \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq 0.965:\\ \;\;\;\;\frac{\sin ky}{t\_3} \cdot th\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin kx) 2.0)))))
        (t_3 (hypot (sin kx) (sin ky))))
   (if (<= t_2 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_2 -0.04)
       (* (/ th t_3) (sin ky))
       (if (<= t_2 0.4)
         (* (/ (sin th) (sqrt (fma (sin kx) (sin kx) (* ky ky)))) (sin ky))
         (if (<= t_2 0.965)
           (* (/ (sin ky) t_3) th)
           (if (<= t_2 1.0)
             (sin th)
             (*
              (/
               1.0
               (/
                (hypot (sin ky) (sin kx))
                (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
              (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(kx), 2.0)));
	double t_3 = hypot(sin(kx), sin(ky));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_2 <= -0.04) {
		tmp = (th / t_3) * sin(ky);
	} else if (t_2 <= 0.4) {
		tmp = (sin(th) / sqrt(fma(sin(kx), sin(kx), (ky * ky)))) * sin(ky);
	} else if (t_2 <= 0.965) {
		tmp = (sin(ky) / t_3) * th;
	} else if (t_2 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (1.0 / (hypot(sin(ky), sin(kx)) / (fma((ky * ky), -0.16666666666666666, 1.0) * ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(kx) ^ 2.0))))
	t_3 = hypot(sin(kx), sin(ky))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_2 <= -0.04)
		tmp = Float64(Float64(th / t_3) * sin(ky));
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(sin(th) / sqrt(fma(sin(kx), sin(kx), Float64(ky * ky)))) * sin(ky));
	elseif (t_2 <= 0.965)
		tmp = Float64(Float64(sin(ky) / t_3) * th);
	elseif (t_2 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(1.0 / Float64(hypot(sin(ky), sin(kx)) / Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky))) * 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[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], 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$2, -0.04], N[(N[(th / t$95$3), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[Sin[kx], $MachinePrecision] * N[Sin[kx], $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.965], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$3), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 1.0], N[Sin[th], $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 82.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0400000000000000008

   1. Initial program 97.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 38.6

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

   5. Applied rewrites 38.6%

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

   7. Applied rewrites 38.7%

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

   if -0.0400000000000000008 < (/.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))))) < 0.40000000000000002

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.4

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 96.4

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

   7. Applied rewrites 96.4%

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

   if 0.40000000000000002 < (/.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))))) < 0.964999999999999969

   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 43.8

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

   5. Applied rewrites 43.8%

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

   7. Applied rewrites 43.9%

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

   if 0.964999999999999969 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.8

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

   5. Applied rewrites 92.8%

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

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

   1. Initial program 2.2%

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

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 2.2

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}} \cdot \sin th \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.04:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.4:\\ \;\;\;\;\frac{\sin th}{\sqrt{\mathsf{fma}\left(\sin kx, \sin kx, ky \cdot ky\right)}} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.965:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin kx}^{2}}}\\ t_3 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.04:\\ \;\;\;\;\frac{th}{t\_3} \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{\sin th}{\sqrt{\mathsf{fma}\left(\sin kx, \sin kx, ky \cdot ky\right)}} \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq 0.965:\\ \;\;\;\;\frac{\sin ky}{t\_3} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin kx) 2.0)))))
        (t_3 (hypot (sin kx) (sin ky))))
   (if (<= t_2 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_2 -0.04)
       (* (/ th t_3) (sin ky))
       (if (<= t_2 0.4)
         (* (/ (sin th) (sqrt (fma (sin kx) (sin kx) (* ky ky)))) (sin ky))
         (if (<= t_2 0.965)
           (* (/ (sin ky) t_3) th)
           (*
            (/ (sin ky) (fma (* 0.5 kx) (/ kx (sin ky)) (sin ky)))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(kx), 2.0)));
	double t_3 = hypot(sin(kx), sin(ky));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_2 <= -0.04) {
		tmp = (th / t_3) * sin(ky);
	} else if (t_2 <= 0.4) {
		tmp = (sin(th) / sqrt(fma(sin(kx), sin(kx), (ky * ky)))) * sin(ky);
	} else if (t_2 <= 0.965) {
		tmp = (sin(ky) / t_3) * th;
	} else {
		tmp = (sin(ky) / fma((0.5 * kx), (kx / sin(ky)), sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(kx) ^ 2.0))))
	t_3 = hypot(sin(kx), sin(ky))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_2 <= -0.04)
		tmp = Float64(Float64(th / t_3) * sin(ky));
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(sin(th) / sqrt(fma(sin(kx), sin(kx), Float64(ky * ky)))) * sin(ky));
	elseif (t_2 <= 0.965)
		tmp = Float64(Float64(sin(ky) / t_3) * th);
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(0.5 * kx), Float64(kx / sin(ky)), sin(ky))) * 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[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], 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$2, -0.04], N[(N[(th / t$95$3), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[Sin[kx], $MachinePrecision] * N[Sin[kx], $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.965], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$3), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(0.5 * kx), $MachinePrecision] * N[(kx / N[Sin[ky], $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 82.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0400000000000000008

   1. Initial program 97.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 38.6

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

   5. Applied rewrites 38.6%

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

   7. Applied rewrites 38.7%

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

   if -0.0400000000000000008 < (/.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))))) < 0.40000000000000002

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.4

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 96.4

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

   7. Applied rewrites 96.4%

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

   if 0.40000000000000002 < (/.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))))) < 0.964999999999999969

   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 43.8

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

   5. Applied rewrites 43.8%

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

   7. Applied rewrites 43.9%

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

   if 0.964999999999999969 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 85.0%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 84.1

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

   5. Applied rewrites 84.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)}} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.04:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.4:\\ \;\;\;\;\frac{\sin th}{\sqrt{\mathsf{fma}\left(\sin kx, \sin kx, ky \cdot ky\right)}} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.965:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin kx}^{2}}}\\ t_3 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.04:\\ \;\;\;\;\frac{th}{t\_3} \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{\sin th}{\sqrt{\mathsf{fma}\left(\sin kx, \sin kx, ky \cdot ky\right)}} \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq 0.965:\\ \;\;\;\;\frac{\sin ky}{t\_3} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin kx) 2.0)))))
        (t_3 (hypot (sin kx) (sin ky))))
   (if (<= t_2 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_2 -0.04)
       (* (/ th t_3) (sin ky))
       (if (<= t_2 0.4)
         (* (/ (sin th) (sqrt (fma (sin kx) (sin kx) (* ky ky)))) (sin ky))
         (if (<= t_2 0.965) (* (/ (sin ky) t_3) th) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(kx), 2.0)));
	double t_3 = hypot(sin(kx), sin(ky));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_2 <= -0.04) {
		tmp = (th / t_3) * sin(ky);
	} else if (t_2 <= 0.4) {
		tmp = (sin(th) / sqrt(fma(sin(kx), sin(kx), (ky * ky)))) * sin(ky);
	} else if (t_2 <= 0.965) {
		tmp = (sin(ky) / t_3) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(kx) ^ 2.0))))
	t_3 = hypot(sin(kx), sin(ky))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_2 <= -0.04)
		tmp = Float64(Float64(th / t_3) * sin(ky));
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(sin(th) / sqrt(fma(sin(kx), sin(kx), Float64(ky * ky)))) * sin(ky));
	elseif (t_2 <= 0.965)
		tmp = Float64(Float64(sin(ky) / t_3) * th);
	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[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], 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$2, -0.04], N[(N[(th / t$95$3), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[Sin[kx], $MachinePrecision] * N[Sin[kx], $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.965], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$3), $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))))) < -1

   1. Initial program 82.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0400000000000000008

   1. Initial program 97.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 38.6

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

   5. Applied rewrites 38.6%

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

   7. Applied rewrites 38.7%

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

   if -0.0400000000000000008 < (/.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))))) < 0.40000000000000002

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.4

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 96.4

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

   7. Applied rewrites 96.4%

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

   if 0.40000000000000002 < (/.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))))) < 0.964999999999999969

   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 43.8

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

   5. Applied rewrites 43.8%

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

   7. Applied rewrites 43.9%

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

   if 0.964999999999999969 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 85.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 84.0

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

   5. Applied rewrites 84.0%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.04:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.4:\\ \;\;\;\;\frac{\sin th}{\sqrt{\mathsf{fma}\left(\sin kx, \sin kx, ky \cdot ky\right)}} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.965:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ t_4 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.04:\\ \;\;\;\;\frac{th}{t\_4} \cdot \sin ky\\ \mathbf{elif}\;t\_3 \leq 0.4:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.965:\\ \;\;\;\;\frac{\sin ky}{t\_4} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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_1 t_2))))
        (t_4 (hypot (sin kx) (sin ky))))
   (if (<= t_3 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_3 -0.04)
       (* (/ th t_4) (sin ky))
       (if (<= t_3 0.4)
         (* (/ (sin ky) (sqrt (+ (* ky ky) t_2))) (sin th))
         (if (<= t_3 0.965) (* (/ (sin ky) t_4) th) (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_1 + t_2));
	double t_4 = hypot(sin(kx), sin(ky));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_3 <= -0.04) {
		tmp = (th / t_4) * sin(ky);
	} else if (t_3 <= 0.4) {
		tmp = (sin(ky) / sqrt(((ky * ky) + t_2))) * sin(th);
	} else if (t_3 <= 0.965) {
		tmp = (sin(ky) / t_4) * 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(ky), 2.0);
	double t_2 = Math.pow(Math.sin(kx), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((t_1 + t_2));
	double t_4 = Math.hypot(Math.sin(kx), Math.sin(ky));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + t_1))) * Math.sin(th);
	} else if (t_3 <= -0.04) {
		tmp = (th / t_4) * Math.sin(ky);
	} else if (t_3 <= 0.4) {
		tmp = (Math.sin(ky) / Math.sqrt(((ky * ky) + t_2))) * Math.sin(th);
	} else if (t_3 <= 0.965) {
		tmp = (Math.sin(ky) / t_4) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = math.sin(ky) / math.sqrt((t_1 + t_2))
	t_4 = math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if t_3 <= -1.0:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + t_1))) * math.sin(th)
	elif t_3 <= -0.04:
		tmp = (th / t_4) * math.sin(ky)
	elif t_3 <= 0.4:
		tmp = (math.sin(ky) / math.sqrt(((ky * ky) + t_2))) * math.sin(th)
	elif t_3 <= 0.965:
		tmp = (math.sin(ky) / t_4) * th
	else:
		tmp = math.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_1 + t_2)))
	t_4 = hypot(sin(kx), sin(ky))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_3 <= -0.04)
		tmp = Float64(Float64(th / t_4) * sin(ky));
	elseif (t_3 <= 0.4)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + t_2))) * sin(th));
	elseif (t_3 <= 0.965)
		tmp = Float64(Float64(sin(ky) / t_4) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(kx) ^ 2.0;
	t_3 = sin(ky) / sqrt((t_1 + t_2));
	t_4 = hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (t_3 <= -1.0)
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	elseif (t_3 <= -0.04)
		tmp = (th / t_4) * sin(ky);
	elseif (t_3 <= 0.4)
		tmp = (sin(ky) / sqrt(((ky * ky) + t_2))) * sin(th);
	elseif (t_3 <= 0.965)
		tmp = (sin(ky) / t_4) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = 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$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], 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, -0.04], N[(N[(th / t$95$4), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.4], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.965], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$4), $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))))) < -1

   1. Initial program 82.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0400000000000000008

   1. Initial program 97.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 38.6

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

   5. Applied rewrites 38.6%

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

   7. Applied rewrites 38.7%

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

   if -0.0400000000000000008 < (/.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))))) < 0.40000000000000002

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if 0.40000000000000002 < (/.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))))) < 0.964999999999999969

   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 43.8

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

   5. Applied rewrites 43.8%

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

   7. Applied rewrites 43.9%

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

   if 0.964999999999999969 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 85.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 84.0

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

   5. Applied rewrites 84.0%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.04:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.4:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + {\sin kx}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.965:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin kx}^{2}}}\\ t_3 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.04:\\ \;\;\;\;\frac{th}{t\_3} \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;t\_2 \leq 0.965:\\ \;\;\;\;\frac{\sin ky}{t\_3} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin kx) 2.0)))))
        (t_3 (hypot (sin kx) (sin ky))))
   (if (<= t_2 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_2 -0.04)
       (* (/ th t_3) (sin ky))
       (if (<= t_2 0.4)
         (/ (* (sin th) ky) (hypot (sin ky) (sin kx)))
         (if (<= t_2 0.965) (* (/ (sin ky) t_3) th) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(kx), 2.0)));
	double t_3 = hypot(sin(kx), sin(ky));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_2 <= -0.04) {
		tmp = (th / t_3) * sin(ky);
	} else if (t_2 <= 0.4) {
		tmp = (sin(th) * ky) / hypot(sin(ky), sin(kx));
	} else if (t_2 <= 0.965) {
		tmp = (sin(ky) / t_3) * 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(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((t_1 + Math.pow(Math.sin(kx), 2.0)));
	double t_3 = Math.hypot(Math.sin(kx), Math.sin(ky));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + t_1))) * Math.sin(th);
	} else if (t_2 <= -0.04) {
		tmp = (th / t_3) * Math.sin(ky);
	} else if (t_2 <= 0.4) {
		tmp = (Math.sin(th) * ky) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else if (t_2 <= 0.965) {
		tmp = (Math.sin(ky) / t_3) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((t_1 + math.pow(math.sin(kx), 2.0)))
	t_3 = math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if t_2 <= -1.0:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + t_1))) * math.sin(th)
	elif t_2 <= -0.04:
		tmp = (th / t_3) * math.sin(ky)
	elif t_2 <= 0.4:
		tmp = (math.sin(th) * ky) / math.hypot(math.sin(ky), math.sin(kx))
	elif t_2 <= 0.965:
		tmp = (math.sin(ky) / t_3) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(kx) ^ 2.0))))
	t_3 = hypot(sin(kx), sin(ky))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_2 <= -0.04)
		tmp = Float64(Float64(th / t_3) * sin(ky));
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(sin(th) * ky) / hypot(sin(ky), sin(kx)));
	elseif (t_2 <= 0.965)
		tmp = Float64(Float64(sin(ky) / t_3) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt((t_1 + (sin(kx) ^ 2.0)));
	t_3 = hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (t_2 <= -1.0)
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	elseif (t_2 <= -0.04)
		tmp = (th / t_3) * sin(ky);
	elseif (t_2 <= 0.4)
		tmp = (sin(th) * ky) / hypot(sin(ky), sin(kx));
	elseif (t_2 <= 0.965)
		tmp = (sin(ky) / t_3) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = 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[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], 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$2, -0.04], N[(N[(th / t$95$3), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.4], 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, 0.965], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$3), $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))))) < -1

   1. Initial program 82.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0400000000000000008

   1. Initial program 97.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 38.6

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

   5. Applied rewrites 38.6%

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

   7. Applied rewrites 38.7%

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

   if -0.0400000000000000008 < (/.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))))) < 0.40000000000000002

   1. Initial program 99.6%

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

      1. lift-*.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 94.9

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

      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 94.9

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

   4. Applied rewrites 94.9%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 91.6

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

   7. Applied rewrites 91.6%

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

   if 0.40000000000000002 < (/.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))))) < 0.964999999999999969

   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 43.8

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

   5. Applied rewrites 43.8%

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

   7. Applied rewrites 43.9%

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

   if 0.964999999999999969 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 85.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 84.0

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

   5. Applied rewrites 84.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.04:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.4:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.965:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-11}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-11)
   (* (/ (sin th) (sin kx)) (sin ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-11) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-11) then
        tmp = (sin(th) / sin(kx)) * sin(ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-11) {
		tmp = (Math.sin(th) / Math.sin(kx)) * Math.sin(ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-11:
		tmp = (math.sin(th) / math.sin(kx)) * math.sin(ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-11)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-11)
		tmp = (sin(th) / sin(kx)) * sin(ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-11], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999939e-12

   1. Initial program 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 94.4

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 33.6

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

   7. Applied rewrites 33.6%

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

   if 9.99999999999999939e-12 < (/.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 89.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.6

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

   5. Applied rewrites 62.6%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-11}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-11}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-11)
   (* (/ (sin ky) (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-11) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-11) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-11) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-11:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-11)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-11)
		tmp = (sin(ky) / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-11], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999939e-12

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 33.6

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

   5. Applied rewrites 33.6%

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

   if 9.99999999999999939e-12 < (/.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 89.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.6

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

   5. Applied rewrites 62.6%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-11}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-11}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-11)
   (/
    1.0
    (/ (/ (sin kx) (sin th)) (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-11) {
		tmp = 1.0 / ((sin(kx) / sin(th)) / (fma((ky * ky), -0.16666666666666666, 1.0) * ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-11)
		tmp = Float64(1.0 / Float64(Float64(sin(kx) / sin(th)) / Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-11], N[(1.0 / N[(N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999939e-12

   1. Initial program 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. 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 91.2

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

      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 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 32.2

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

   7. Applied rewrites 32.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 33.6

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

   9. Applied rewrites 33.6%

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

   10. Taylor expanded in ky around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 31.5

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

   12. Applied rewrites 31.5%

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

   if 9.99999999999999939e-12 < (/.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 89.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.6

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

   5. Applied rewrites 62.6%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-11}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-11}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-11)
   (/ (sin th) (/ (sin kx) ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-11) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-11) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-11) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-11:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-11)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-11)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-11], 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))))) < 9.99999999999999939e-12

   1. Initial program 94.3%

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

      1. lift-*.f64 N/A

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

      2. *-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 94.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 32.0

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

   7. Applied rewrites 32.0%

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

   if 9.99999999999999939e-12 < (/.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 89.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.6

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

   5. Applied rewrites 62.6%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-11}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-11}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-11)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-11) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-11) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-11) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-11:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-11)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-11)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-11], 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))))) < 9.99999999999999939e-12

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 32.0

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

   5. Applied rewrites 32.0%

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

   if 9.99999999999999939e-12 < (/.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 89.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.6

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

   5. Applied rewrites 62.6%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-11}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;{\left({\sin ky}^{2}\right)}^{-0.5} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;\sin ky \leq 10^{-169}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (* (pow (pow (sin ky) 2.0) -0.5) (* th (sin ky)))
   (if (<= (sin ky) 1e-169)
     (* (/ (sin th) (sin kx)) (sin ky))
     (if (<= (sin ky) 5e-10)
       (*
        (/ (sin ky) (sqrt (+ (- 0.5 (* 0.5 (cos (* 2.0 kx)))) (* ky ky))))
        (sin th))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = pow(pow(sin(ky), 2.0), -0.5) * (th * sin(ky));
	} else if (sin(ky) <= 1e-169) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} else if (sin(ky) <= 5e-10) {
		tmp = (sin(ky) / sqrt(((0.5 - (0.5 * cos((2.0 * kx)))) + (ky * ky)))) * 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) <= (-0.02d0)) then
        tmp = ((sin(ky) ** 2.0d0) ** (-0.5d0)) * (th * sin(ky))
    else if (sin(ky) <= 1d-169) then
        tmp = (sin(th) / sin(kx)) * sin(ky)
    else if (sin(ky) <= 5d-10) then
        tmp = (sin(ky) / sqrt(((0.5d0 - (0.5d0 * cos((2.0d0 * kx)))) + (ky * ky)))) * 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) <= -0.02) {
		tmp = Math.pow(Math.pow(Math.sin(ky), 2.0), -0.5) * (th * Math.sin(ky));
	} else if (Math.sin(ky) <= 1e-169) {
		tmp = (Math.sin(th) / Math.sin(kx)) * Math.sin(ky);
	} else if (Math.sin(ky) <= 5e-10) {
		tmp = (Math.sin(ky) / Math.sqrt(((0.5 - (0.5 * Math.cos((2.0 * kx)))) + (ky * ky)))) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = math.pow(math.pow(math.sin(ky), 2.0), -0.5) * (th * math.sin(ky))
	elif math.sin(ky) <= 1e-169:
		tmp = (math.sin(th) / math.sin(kx)) * math.sin(ky)
	elif math.sin(ky) <= 5e-10:
		tmp = (math.sin(ky) / math.sqrt(((0.5 - (0.5 * math.cos((2.0 * kx)))) + (ky * ky)))) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(((sin(ky) ^ 2.0) ^ -0.5) * Float64(th * sin(ky)));
	elseif (sin(ky) <= 1e-169)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	elseif (sin(ky) <= 5e-10)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx)))) + Float64(ky * ky)))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = ((sin(ky) ^ 2.0) ^ -0.5) * (th * sin(ky));
	elseif (sin(ky) <= 1e-169)
		tmp = (sin(th) / sin(kx)) * sin(ky);
	elseif (sin(ky) <= 5e-10)
		tmp = (sin(ky) / sqrt(((0.5 - (0.5 * cos((2.0 * kx)))) + (ky * ky)))) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(N[Power[N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision], -0.5], $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-169], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-10], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.0200000000000000004

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 48.0

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

   5. Applied rewrites 48.0%

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 3.3%

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

   12. Applied rewrites 30.4%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 1.00000000000000002e-169

   1. Initial program 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 81.5

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 47.4

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

   7. Applied rewrites 47.4%

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

   if 1.00000000000000002e-169 < (sin.f64 ky) < 5.00000000000000031e-10

   1. Initial program 99.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. lower-*.f64 93.7

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

   7. Applied rewrites 93.7%

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

   if 5.00000000000000031e-10 < (sin.f64 ky)

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.6

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

   5. Applied rewrites 62.6%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;{\left({\sin ky}^{2}\right)}^{-0.5} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;\sin ky \leq 10^{-169}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-11}:\\ \;\;\;\;\frac{th}{\sin kx} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-11)
   (* (/ th (sin kx)) ky)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-11) {
		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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-11) 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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-11) {
		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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-11:
		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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-11)
		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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-11)
		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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-11], 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))))) < 9.99999999999999939e-12

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 20.9%

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

   if 9.99999999999999939e-12 < (/.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 89.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.6

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

   5. Applied rewrites 62.6%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-11}:\\ \;\;\;\;\frac{th}{\sin kx} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 1.75 \cdot 10^{-117}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))
      1.75e-117)
   0.0
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1.75e-117) {
		tmp = 0.0;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1.75d-117) then
        tmp = 0.0d0
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1.75e-117) {
		tmp = 0.0;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1.75e-117:
		tmp = 0.0
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1.75e-117)
		tmp = 0.0;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1.75e-117)
		tmp = 0.0;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.75e-117], 0.0, N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.7499999999999999e-117

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 92.2

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

      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 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

   9. Applied rewrites 17.0%

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

   10. Taylor expanded in th around 0

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

       1. cos-neg N/A

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

       2. +-inverses 16.8

          \[\leadsto \color{blue}{0} \]

   12. Applied rewrites 16.8%

       \[\leadsto \color{blue}{0} \]

   if 1.7499999999999999e-117 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 53.7

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

   5. Applied rewrites 53.7%

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

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 1.75 \cdot 10^{-117}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-169}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-169)
   (* (/ (sin th) (sin kx)) (sin ky))
   (if (<= (sin ky) 5e-10)
     (*
      (/ (sin ky) (sqrt (+ (- 0.5 (* 0.5 (cos (* 2.0 kx)))) (* ky ky))))
      (sin th))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-169) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} else if (sin(ky) <= 5e-10) {
		tmp = (sin(ky) / sqrt(((0.5 - (0.5 * cos((2.0 * kx)))) + (ky * ky)))) * 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) <= 1d-169) then
        tmp = (sin(th) / sin(kx)) * sin(ky)
    else if (sin(ky) <= 5d-10) then
        tmp = (sin(ky) / sqrt(((0.5d0 - (0.5d0 * cos((2.0d0 * kx)))) + (ky * ky)))) * 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) <= 1e-169) {
		tmp = (Math.sin(th) / Math.sin(kx)) * Math.sin(ky);
	} else if (Math.sin(ky) <= 5e-10) {
		tmp = (Math.sin(ky) / Math.sqrt(((0.5 - (0.5 * Math.cos((2.0 * kx)))) + (ky * ky)))) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-169:
		tmp = (math.sin(th) / math.sin(kx)) * math.sin(ky)
	elif math.sin(ky) <= 5e-10:
		tmp = (math.sin(ky) / math.sqrt(((0.5 - (0.5 * math.cos((2.0 * kx)))) + (ky * ky)))) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-169)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	elseif (sin(ky) <= 5e-10)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx)))) + Float64(ky * ky)))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-169)
		tmp = (sin(th) / sin(kx)) * sin(ky);
	elseif (sin(ky) <= 5e-10)
		tmp = (sin(ky) / sqrt(((0.5 - (0.5 * cos((2.0 * kx)))) + (ky * ky)))) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-169], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-10], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < 1.00000000000000002e-169

   1. Initial program 88.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.6

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 31.8

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

   7. Applied rewrites 31.8%

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

   if 1.00000000000000002e-169 < (sin.f64 ky) < 5.00000000000000031e-10

   1. Initial program 99.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. lower-*.f64 93.7

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

   7. Applied rewrites 93.7%

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

   if 5.00000000000000031e-10 < (sin.f64 ky)

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.6

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

   5. Applied rewrites 62.6%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-169}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 20.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 6 \cdot 10^{-64}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 6e-64)
   0.0
   (*
    (fma
     (fma 0.008333333333333333 (* th th) -0.16666666666666666)
     (* th th)
     1.0)
    th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 6e-64) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(0.008333333333333333, (th * th), -0.16666666666666666), (th * th), 1.0) * th;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 6e-64)
		tmp = 0.0;
	else
		tmp = Float64(fma(fma(0.008333333333333333, Float64(th * th), -0.16666666666666666), Float64(th * th), 1.0) * th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 6e-64], 0.0, N[(N[(N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * 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))))) < 6.0000000000000001e-64

   1. Initial program 94.0%

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

      1. lift-*.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 91.3

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

      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 95.7

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

   9. Applied rewrites 16.5%

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

   10. Taylor expanded in th around 0

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

       1. cos-neg N/A

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

       2. +-inverses 16.2

          \[\leadsto \color{blue}{0} \]

   12. Applied rewrites 16.2%

       \[\leadsto \color{blue}{0} \]

   if 6.0000000000000001e-64 < (/.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 90.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 57.1

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

   5. Applied rewrites 57.1%

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

   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 26.9%

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

   10. Applied rewrites 26.9%

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

   11. Taylor expanded in th around 0

       \[\leadsto th \cdot \color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 27.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th \cdot th, 1\right) \cdot \color{blue}{th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 6 \cdot 10^{-64}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 21.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 4 \cdot 10^{-49}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(th \cdot th\right) \cdot -0.16666666666666666, th, th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 4e-49)
   0.0
   (fma (* (* th th) -0.16666666666666666) th th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 4e-49) {
		tmp = 0.0;
	} else {
		tmp = fma(((th * th) * -0.16666666666666666), th, th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 4e-49)
		tmp = 0.0;
	else
		tmp = fma(Float64(Float64(th * th) * -0.16666666666666666), th, th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-49], 0.0, N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th + 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))))) < 3.99999999999999975e-49

   1. Initial program 94.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 90.9

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

      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 95.2

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 32.7

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

   7. Applied rewrites 32.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

   9. Applied rewrites 16.2%

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

   10. Taylor expanded in th around 0

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

       1. cos-neg N/A

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

       2. +-inverses 16.0

          \[\leadsto \color{blue}{0} \]

   12. Applied rewrites 16.0%

       \[\leadsto \color{blue}{0} \]

   if 3.99999999999999975e-49 < (/.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 90.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 59.2

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

   5. Applied rewrites 59.2%

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

   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 27.8%

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

   10. Applied rewrites 27.8%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 4 \cdot 10^{-49}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(th \cdot th\right) \cdot -0.16666666666666666, th, th\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0013:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, \left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2\right)}}{2}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0013)
   (*
    (/
     1.0
     (/
      (hypot (sin ky) (sin kx))
      (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
    (sin th))
   (/
    (sin th)
    (/
     (/
      (sqrt
       (fma (- 1.0 (cos (* 2.0 ky))) 2.0 (* (- 1.0 (cos (* 2.0 kx))) 2.0)))
      2.0)
     (sin ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0013) {
		tmp = (1.0 / (hypot(sin(ky), sin(kx)) / (fma((ky * ky), -0.16666666666666666, 1.0) * ky))) * sin(th);
	} else {
		tmp = sin(th) / ((sqrt(fma((1.0 - cos((2.0 * ky))), 2.0, ((1.0 - cos((2.0 * kx))) * 2.0))) / 2.0) / sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0013)
		tmp = Float64(Float64(1.0 / Float64(hypot(sin(ky), sin(kx)) / Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky))) * sin(th));
	else
		tmp = Float64(sin(th) / Float64(Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 2.0, Float64(Float64(1.0 - cos(Float64(2.0 * kx))) * 2.0))) / 2.0) / sin(ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0013], N[(N[(1.0 / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.0012999999999999999

   1. Initial program 90.4%

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

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 90.3

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 65.9

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

   7. Applied rewrites 65.9%

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

   if 0.0012999999999999999 < ky

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.7

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. sin-mult N/A

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

      8. frac-add N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}}{\sin ky}} \]

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{\color{blue}{4}}}}{\sin ky}} \]

      13. metadata-eval N/A

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

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}}{\sin ky}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0013:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, \left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2\right)}}{2}}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0013:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, \left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0013)
   (*
    (/
     1.0
     (/
      (hypot (sin ky) (sin kx))
      (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
    (sin th))
   (*
    (/
     (sin ky)
     (/
      (sqrt
       (fma (- 1.0 (cos (* 2.0 ky))) 2.0 (* (- 1.0 (cos (* 2.0 kx))) 2.0)))
      2.0))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0013) {
		tmp = (1.0 / (hypot(sin(ky), sin(kx)) / (fma((ky * ky), -0.16666666666666666, 1.0) * ky))) * sin(th);
	} else {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((2.0 * ky))), 2.0, ((1.0 - cos((2.0 * kx))) * 2.0))) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0013)
		tmp = Float64(Float64(1.0 / Float64(hypot(sin(ky), sin(kx)) / Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 2.0, Float64(Float64(1.0 - cos(Float64(2.0 * kx))) * 2.0))) / 2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0013], N[(N[(1.0 / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.0012999999999999999

   1. Initial program 90.4%

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

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 90.3

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 65.9

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

   7. Applied rewrites 65.9%

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

   if 0.0012999999999999999 < ky

   1. Initial program 99.6%

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

      1. lift-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 99.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0013:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, \left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2\right)}}{2}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 62.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.004:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.004)
   (* (/ (sin ky) (hypot (sin kx) (sin ky))) th)
   (/ (* (sin th) ky) (hypot (sin ky) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.004) {
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	} else {
		tmp = (sin(th) * ky) / hypot(sin(ky), sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.004) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky))) * th;
	} else {
		tmp = (Math.sin(th) * ky) / Math.hypot(Math.sin(ky), Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.004:
		tmp = (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky))) * th
	else:
		tmp = (math.sin(th) * ky) / math.hypot(math.sin(ky), math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.004)
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * th);
	else
		tmp = Float64(Float64(sin(th) * ky) / hypot(sin(ky), sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.004)
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	else
		tmp = (sin(th) * ky) / hypot(sin(ky), sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.004], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.0040000000000000001

   1. Initial program 92.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 56.8

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

   5. Applied rewrites 56.8%

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

   7. Applied rewrites 64.5%

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

   if 0.0040000000000000001 < th

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 92.6

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

      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 99.5

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

   4. Applied rewrites 99.5%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 49.6

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

   7. Applied rewrites 49.6%

      \[\leadsto \frac{\color{blue}{ky \cdot \sin th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.004:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 58.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.235:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + ky \cdot ky}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.235)
   (* (/ (sin ky) (hypot (sin kx) (sin ky))) th)
   (*
    (/ (sin ky) (sqrt (+ (- 0.5 (* 0.5 (cos (* 2.0 kx)))) (* ky ky))))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.235) {
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	} else {
		tmp = (sin(ky) / sqrt(((0.5 - (0.5 * cos((2.0 * kx)))) + (ky * ky)))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.235) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky))) * th;
	} else {
		tmp = (Math.sin(ky) / Math.sqrt(((0.5 - (0.5 * Math.cos((2.0 * kx)))) + (ky * ky)))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.235:
		tmp = (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky))) * th
	else:
		tmp = (math.sin(ky) / math.sqrt(((0.5 - (0.5 * math.cos((2.0 * kx)))) + (ky * ky)))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.235)
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * th);
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx)))) + Float64(ky * ky)))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.235)
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	else
		tmp = (sin(ky) / sqrt(((0.5 - (0.5 * cos((2.0 * kx)))) + (ky * ky)))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.235], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.23499999999999999

   1. Initial program 92.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 56.8

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

   5. Applied rewrites 56.8%

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

   7. Applied rewrites 64.5%

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

   if 0.23499999999999999 < th

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 43.7

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

   5. Applied rewrites 43.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. lower-*.f64 38.2

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

   7. Applied rewrites 38.2%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.235:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + ky \cdot ky}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 58.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.235:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + ky \cdot ky}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.235)
   (* (/ th (hypot (sin kx) (sin ky))) (sin ky))
   (*
    (/ (sin ky) (sqrt (+ (- 0.5 (* 0.5 (cos (* 2.0 kx)))) (* ky ky))))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.235) {
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	} else {
		tmp = (sin(ky) / sqrt(((0.5 - (0.5 * cos((2.0 * kx)))) + (ky * ky)))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.235) {
		tmp = (th / Math.hypot(Math.sin(kx), Math.sin(ky))) * Math.sin(ky);
	} else {
		tmp = (Math.sin(ky) / Math.sqrt(((0.5 - (0.5 * Math.cos((2.0 * kx)))) + (ky * ky)))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.235:
		tmp = (th / math.hypot(math.sin(kx), math.sin(ky))) * math.sin(ky)
	else:
		tmp = (math.sin(ky) / math.sqrt(((0.5 - (0.5 * math.cos((2.0 * kx)))) + (ky * ky)))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.235)
		tmp = Float64(Float64(th / hypot(sin(kx), sin(ky))) * sin(ky));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx)))) + Float64(ky * ky)))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.235)
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	else
		tmp = (sin(ky) / sqrt(((0.5 - (0.5 * cos((2.0 * kx)))) + (ky * ky)))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.235], N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.23499999999999999

   1. Initial program 92.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 56.8

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

   5. Applied rewrites 56.8%

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

   7. Applied rewrites 64.4%

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

   if 0.23499999999999999 < th

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 43.7

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

   5. Applied rewrites 43.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. lower-*.f64 38.2

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

   7. Applied rewrites 38.2%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.235:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + ky \cdot ky}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 11.9% accurate, 632.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 0.0)

C

double code(double kx, double ky, double th) {
	return 0.0;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = 0.0d0
end function

Java

public static double code(double kx, double ky, double th) {
	return 0.0;
}

Python

def code(kx, ky, th):
	return 0.0

Julia

function code(kx, ky, th)
	return 0.0
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = 0.0;
end

Wolfram

code[kx_, ky_, th_] := 0.0

Derivation

1. Initial program 92.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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 90.3

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

   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 94.7

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

4. Applied rewrites 94.7%

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

5. Taylor expanded in ky around 0

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

   1. lower-sin.f64 24.7

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

7. Applied rewrites 24.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. sin-mult N/A

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

   6. div-sub N/A

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

   7. div-sub N/A

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

   8. lower--.f64 N/A

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

9. Applied rewrites 12.5%

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

10. Taylor expanded in th around 0

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

    1. cos-neg N/A

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

    2. +-inverses 12.1

       \[\leadsto \color{blue}{0} \]

12. Applied rewrites 12.1%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024273 
(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)))