Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 13.8s

Alternatives: 20

Speedup: 1.4×

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, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

3. Taylor expanded in kx around inf

   \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   6. sin-lowering-sin.f64 99.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

5. Simplified 99.7%

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

Alternative 2: 42.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \frac{1 + -0.16666666666666666 \cdot \left(th \cdot th\right)}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\sin th \cdot \left(1 + \frac{\frac{-0.5 \cdot \left(kx \cdot kx\right)}{ky}}{ky}\right)\\ \mathbf{elif}\;\sin kx \leq 10^{-50}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.04)
   (*
    (* (sin ky) th)
    (/ (+ 1.0 (* -0.16666666666666666 (* th th))) (hypot ky (sin kx))))
   (if (<= (sin kx) 2e-117)
     (* (sin th) (+ 1.0 (/ (/ (* -0.5 (* kx kx)) ky) ky)))
     (if (<= (sin kx) 1e-50)
       (* th (/ (sin ky) (hypot kx (sin ky))))
       (* (sin ky) (/ (sin th) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.04) {
		tmp = (sin(ky) * th) * ((1.0 + (-0.16666666666666666 * (th * th))) / hypot(ky, sin(kx)));
	} else if (sin(kx) <= 2e-117) {
		tmp = sin(th) * (1.0 + (((-0.5 * (kx * kx)) / ky) / ky));
	} else if (sin(kx) <= 1e-50) {
		tmp = th * (sin(ky) / hypot(kx, sin(ky)));
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.04) {
		tmp = (Math.sin(ky) * th) * ((1.0 + (-0.16666666666666666 * (th * th))) / Math.hypot(ky, Math.sin(kx)));
	} else if (Math.sin(kx) <= 2e-117) {
		tmp = Math.sin(th) * (1.0 + (((-0.5 * (kx * kx)) / ky) / ky));
	} else if (Math.sin(kx) <= 1e-50) {
		tmp = th * (Math.sin(ky) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.04:
		tmp = (math.sin(ky) * th) * ((1.0 + (-0.16666666666666666 * (th * th))) / math.hypot(ky, math.sin(kx)))
	elif math.sin(kx) <= 2e-117:
		tmp = math.sin(th) * (1.0 + (((-0.5 * (kx * kx)) / ky) / ky))
	elif math.sin(kx) <= 1e-50:
		tmp = th * (math.sin(ky) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.04)
		tmp = Float64(Float64(sin(ky) * th) * Float64(Float64(1.0 + Float64(-0.16666666666666666 * Float64(th * th))) / hypot(ky, sin(kx))));
	elseif (sin(kx) <= 2e-117)
		tmp = Float64(sin(th) * Float64(1.0 + Float64(Float64(Float64(-0.5 * Float64(kx * kx)) / ky) / ky)));
	elseif (sin(kx) <= 1e-50)
		tmp = Float64(th * Float64(sin(ky) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.04)
		tmp = (sin(ky) * th) * ((1.0 + (-0.16666666666666666 * (th * th))) / hypot(ky, sin(kx)));
	elseif (sin(kx) <= 2e-117)
		tmp = sin(th) * (1.0 + (((-0.5 * (kx * kx)) / ky) / ky));
	elseif (sin(kx) <= 1e-50)
		tmp = th * (sin(ky) / hypot(kx, sin(ky)));
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.04], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[(N[(1.0 + N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-117], N[(N[Sin[th], $MachinePrecision] * N[(1.0 + N[(N[(N[(-0.5 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] / ky), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-50], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.0400000000000000008

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right), \mathsf{hypot.f64}\left(\color{blue}{\mathsf{sin.f64}\left(ky\right)}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {th}^{2}\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(th \cdot th\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot th\right) \cdot th\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot th\right), th\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      6. *-lowering-*.f64 46.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, th\right), th\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   7. Simplified 46.2%

      \[\leadsto \sin ky \cdot \frac{\color{blue}{th \cdot \left(1 + \left(-0.16666666666666666 \cdot th\right) \cdot th\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sin ky \cdot th\right), \color{blue}{\left(\frac{1 + \left(\frac{-1}{6} \cdot th\right) \cdot th}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin ky, th\right), \left(\frac{\color{blue}{1 + \left(\frac{-1}{6} \cdot th\right) \cdot th}}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), th\right), \left(\frac{\color{blue}{1} + \left(\frac{-1}{6} \cdot th\right) \cdot th}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), th\right), \mathsf{/.f64}\left(\left(1 + \left(\frac{-1}{6} \cdot th\right) \cdot th\right), \color{blue}{\left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot th\right) \cdot th\right)\right), \left(\sqrt{\color{blue}{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(th \cdot th\right)\right)\right), \left(\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(th \cdot th\right)\right)\right), \left(\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, th\right)\right)\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \color{blue}{\sin kx}}\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, th\right)\right)\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, th\right)\right)\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, th\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      14. sin-lowering-sin.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, th\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   9. Applied egg-rr 46.1%

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

   10. Taylor expanded in ky around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, th\right)\right)\right), \mathsf{hypot.f64}\left(\color{blue}{ky}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 28.9%

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

   if -0.0400000000000000008 < (sin.f64 kx) < 2.00000000000000006e-117

   1. Initial program 85.7%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}\right)}, \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(\color{blue}{th}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot {kx}^{2}}{{\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot {kx}^{2}\right), \left({\sin ky}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left({kx}^{2}\right)\right), \left({\sin ky}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(kx \cdot kx\right)\right), \left({\sin ky}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(kx, kx\right)\right), \left({\sin ky}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(kx, kx\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      8. sin-lowering-sin.f64 40.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(kx, kx\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 40.1%

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

   6. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{{kx}^{2}}{{ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot {kx}^{2}}{{ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot {kx}^{2}}{ky \cdot ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\frac{-1}{2} \cdot {kx}^{2}}{ky}}{ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{-1}{2} \cdot {kx}^{2}}{ky}\right), ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot {kx}^{2}\right), ky\right), ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left({kx}^{2}\right)\right), ky\right), ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(kx \cdot kx\right)\right), ky\right), ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      8. *-lowering-*.f64 45.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(kx, kx\right)\right), ky\right), ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   8. Simplified 45.7%

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

   if 2.00000000000000006e-117 < (sin.f64 kx) < 1.00000000000000001e-50

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.8%

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

   6. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\color{blue}{kx}, \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 99.8%

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

   9. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(kx, \mathsf{sin.f64}\left(ky\right)\right)\right), \color{blue}{th}\right) \]
   10. Step-by-step derivation

   11. Simplified 64.1%

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

   if 1.00000000000000001e-50 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\frac{\sin th}{\sin kx}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\sin kx}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \sin \color{blue}{kx}\right)\right) \]

      3. sin-lowering-sin.f64 63.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{sin.f64}\left(kx\right)\right)\right) \]

   7. Simplified 63.8%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \frac{1 + -0.16666666666666666 \cdot \left(th \cdot th\right)}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\sin th \cdot \left(1 + \frac{\frac{-0.5 \cdot \left(kx \cdot kx\right)}{ky}}{ky}\right)\\ \mathbf{elif}\;\sin kx \leq 10^{-50}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\sin ky \cdot \frac{th \cdot \left(1 + th \cdot \left(th \cdot -0.16666666666666666\right)\right)}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\sin th \cdot \left(1 + \frac{\frac{-0.5 \cdot \left(kx \cdot kx\right)}{ky}}{ky}\right)\\ \mathbf{elif}\;\sin kx \leq 10^{-50}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.04)
   (*
    (sin ky)
    (/ (* th (+ 1.0 (* th (* th -0.16666666666666666)))) (hypot ky (sin kx))))
   (if (<= (sin kx) 2e-117)
     (* (sin th) (+ 1.0 (/ (/ (* -0.5 (* kx kx)) ky) ky)))
     (if (<= (sin kx) 1e-50)
       (* th (/ (sin ky) (hypot kx (sin ky))))
       (* (sin ky) (/ (sin th) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.04) {
		tmp = sin(ky) * ((th * (1.0 + (th * (th * -0.16666666666666666)))) / hypot(ky, sin(kx)));
	} else if (sin(kx) <= 2e-117) {
		tmp = sin(th) * (1.0 + (((-0.5 * (kx * kx)) / ky) / ky));
	} else if (sin(kx) <= 1e-50) {
		tmp = th * (sin(ky) / hypot(kx, sin(ky)));
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.04) {
		tmp = Math.sin(ky) * ((th * (1.0 + (th * (th * -0.16666666666666666)))) / Math.hypot(ky, Math.sin(kx)));
	} else if (Math.sin(kx) <= 2e-117) {
		tmp = Math.sin(th) * (1.0 + (((-0.5 * (kx * kx)) / ky) / ky));
	} else if (Math.sin(kx) <= 1e-50) {
		tmp = th * (Math.sin(ky) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.04:
		tmp = math.sin(ky) * ((th * (1.0 + (th * (th * -0.16666666666666666)))) / math.hypot(ky, math.sin(kx)))
	elif math.sin(kx) <= 2e-117:
		tmp = math.sin(th) * (1.0 + (((-0.5 * (kx * kx)) / ky) / ky))
	elif math.sin(kx) <= 1e-50:
		tmp = th * (math.sin(ky) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.04)
		tmp = Float64(sin(ky) * Float64(Float64(th * Float64(1.0 + Float64(th * Float64(th * -0.16666666666666666)))) / hypot(ky, sin(kx))));
	elseif (sin(kx) <= 2e-117)
		tmp = Float64(sin(th) * Float64(1.0 + Float64(Float64(Float64(-0.5 * Float64(kx * kx)) / ky) / ky)));
	elseif (sin(kx) <= 1e-50)
		tmp = Float64(th * Float64(sin(ky) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.04)
		tmp = sin(ky) * ((th * (1.0 + (th * (th * -0.16666666666666666)))) / hypot(ky, sin(kx)));
	elseif (sin(kx) <= 2e-117)
		tmp = sin(th) * (1.0 + (((-0.5 * (kx * kx)) / ky) / ky));
	elseif (sin(kx) <= 1e-50)
		tmp = th * (sin(ky) / hypot(kx, sin(ky)));
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.04], N[(N[Sin[ky], $MachinePrecision] * N[(N[(th * N[(1.0 + N[(th * N[(th * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-117], N[(N[Sin[th], $MachinePrecision] * N[(1.0 + N[(N[(N[(-0.5 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] / ky), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-50], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.0400000000000000008

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right), \mathsf{hypot.f64}\left(\color{blue}{\mathsf{sin.f64}\left(ky\right)}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {th}^{2}\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(th \cdot th\right)\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot th\right) \cdot th\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot th\right), th\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

      6. *-lowering-*.f64 46.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, th\right), th\right)\right)\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   7. Simplified 46.2%

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

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, th\right), th\right)\right)\right), \mathsf{hypot.f64}\left(\color{blue}{ky}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 28.9%

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

   if -0.0400000000000000008 < (sin.f64 kx) < 2.00000000000000006e-117

   1. Initial program 85.7%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}\right)}, \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(\color{blue}{th}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot {kx}^{2}}{{\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot {kx}^{2}\right), \left({\sin ky}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left({kx}^{2}\right)\right), \left({\sin ky}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(kx \cdot kx\right)\right), \left({\sin ky}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(kx, kx\right)\right), \left({\sin ky}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(kx, kx\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      8. sin-lowering-sin.f64 40.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(kx, kx\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 40.1%

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

   6. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{{kx}^{2}}{{ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot {kx}^{2}}{{ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot {kx}^{2}}{ky \cdot ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\frac{-1}{2} \cdot {kx}^{2}}{ky}}{ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{-1}{2} \cdot {kx}^{2}}{ky}\right), ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot {kx}^{2}\right), ky\right), ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left({kx}^{2}\right)\right), ky\right), ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(kx \cdot kx\right)\right), ky\right), ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      8. *-lowering-*.f64 45.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(kx, kx\right)\right), ky\right), ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   8. Simplified 45.7%

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

   if 2.00000000000000006e-117 < (sin.f64 kx) < 1.00000000000000001e-50

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.8%

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

   6. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\color{blue}{kx}, \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 99.8%

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

   9. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(kx, \mathsf{sin.f64}\left(ky\right)\right)\right), \color{blue}{th}\right) \]
   10. Step-by-step derivation

   11. Simplified 64.1%

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

   if 1.00000000000000001e-50 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\frac{\sin th}{\sin kx}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\sin kx}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \sin \color{blue}{kx}\right)\right) \]

      3. sin-lowering-sin.f64 63.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{sin.f64}\left(kx\right)\right)\right) \]

   7. Simplified 63.8%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\sin ky \cdot \frac{th \cdot \left(1 + th \cdot \left(th \cdot -0.16666666666666666\right)\right)}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\sin th \cdot \left(1 + \frac{\frac{-0.5 \cdot \left(kx \cdot kx\right)}{ky}}{ky}\right)\\ \mathbf{elif}\;\sin kx \leq 10^{-50}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 0.0001:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.01)
   (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) 0.0001)
     (* (sin ky) (/ (sin th) (hypot ky (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else if (sin(ky) <= 0.0001) {
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.01) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (Math.sin(ky) <= 0.0001) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.01:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	elif math.sin(ky) <= 0.0001:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.01)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	elseif (sin(ky) <= 0.0001)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(ky, sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.01)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	elseif (sin(ky) <= 0.0001)
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.01], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.0001], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0100000000000000002

   1. Initial program 99.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{th}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 48.0%

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

   if -0.0100000000000000002 < (sin.f64 ky) < 1.00000000000000005e-4

   1. Initial program 89.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\color{blue}{ky}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 99.6%

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

   if 1.00000000000000005e-4 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 61.5%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 61.5%

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

Alternative 5: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

   10. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

   13. sin-lowering-sin.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

3. Simplified 99.6%

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

Alternative 6: 64.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0136:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{elif}\;ky \leq 8.8 \cdot 10^{+163}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0136)
   (* (sin th) (/ (sin ky) (hypot (sin kx) ky)))
   (if (<= ky 8.8e+163)
     (* (sin th) (/ (sin ky) (hypot kx (sin ky))))
     (* (sin ky) (/ th (hypot (sin ky) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0136) {
		tmp = sin(th) * (sin(ky) / hypot(sin(kx), ky));
	} else if (ky <= 8.8e+163) {
		tmp = sin(th) * (sin(ky) / hypot(kx, sin(ky)));
	} else {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0136) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(kx), ky));
	} else if (ky <= 8.8e+163) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.0136:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(kx), ky))
	elif ky <= 8.8e+163:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0136)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(kx), ky)));
	elseif (ky <= 8.8e+163)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.0136)
		tmp = sin(th) * (sin(ky) / hypot(sin(kx), ky));
	elseif (ky <= 8.8e+163)
		tmp = sin(th) * (sin(ky) / hypot(kx, sin(ky)));
	else
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0136], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 8.8e+163], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 0.0135999999999999992

   1. Initial program 92.2%

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

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.7%

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

   6. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \color{blue}{ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 71.8%

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

   if 0.0135999999999999992 < ky < 8.79999999999999945e163

   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 inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.9%

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

   6. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\color{blue}{kx}, \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 56.4%

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

   if 8.79999999999999945e163 < ky

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{th}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 57.2%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0136:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{elif}\;ky \leq 8.8 \cdot 10^{+163}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0075:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;ky \leq 5.6 \cdot 10^{+164}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0075)
   (* (sin ky) (/ (sin th) (hypot ky (sin kx))))
   (if (<= ky 5.6e+164)
     (* (sin th) (/ (sin ky) (hypot kx (sin ky))))
     (* (sin ky) (/ th (hypot (sin ky) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0075) {
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	} else if (ky <= 5.6e+164) {
		tmp = sin(th) * (sin(ky) / hypot(kx, sin(ky)));
	} else {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0075) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else if (ky <= 5.6e+164) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.0075:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	elif ky <= 5.6e+164:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0075)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(ky, sin(kx))));
	elseif (ky <= 5.6e+164)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.0075)
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	elseif (ky <= 5.6e+164)
		tmp = sin(th) * (sin(ky) / hypot(kx, sin(ky)));
	else
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0075], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 5.6e+164], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 0.0074999999999999997

   1. Initial program 92.2%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\color{blue}{ky}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 71.7%

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

   if 0.0074999999999999997 < ky < 5.6000000000000004e164

   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 inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.9%

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

   6. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\color{blue}{kx}, \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 56.4%

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

   if 5.6000000000000004e164 < ky

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{th}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 57.2%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0075:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;ky \leq 5.6 \cdot 10^{+164}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.00195:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;ky \leq 4.8 \cdot 10^{+162}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.00195)
   (* (sin ky) (/ (sin th) (hypot ky (sin kx))))
   (if (<= ky 4.8e+162)
     (* (sin ky) (/ (sin th) (hypot (sin ky) kx)))
     (* (sin ky) (/ th (hypot (sin ky) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.00195) {
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	} else if (ky <= 4.8e+162) {
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	} else {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.00195) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else if (ky <= 4.8e+162) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(Math.sin(ky), kx));
	} else {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.00195:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	elif ky <= 4.8e+162:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(math.sin(ky), kx))
	else:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.00195)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(ky, sin(kx))));
	elseif (ky <= 4.8e+162)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(sin(ky), kx)));
	else
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.00195)
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	elseif (ky <= 4.8e+162)
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	else
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.00195], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 4.8e+162], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 0.0019499999999999999

   1. Initial program 92.2%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\color{blue}{ky}, \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 71.7%

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

   if 0.0019499999999999999 < ky < 4.80000000000000018e162

   1. Initial program 99.9%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{kx}\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 54.7%

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

   if 4.80000000000000018e162 < ky

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{th}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 58.7%

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

Alternative 9: 56.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 380000:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 1.08 \cdot 10^{+275}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 380000.0)
   (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
   (if (<= th 1.08e+275) (sin th) (* (sin ky) (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 380000.0) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else if (th <= 1.08e+275) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 380000.0) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (th <= 1.08e+275) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 380000.0:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	elif th <= 1.08e+275:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 380000.0)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	elseif (th <= 1.08e+275)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 380000.0)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	elseif (th <= 1.08e+275)
		tmp = sin(th);
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 380000.0], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 1.08e+275], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 3.8e5

   1. Initial program 93.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\color{blue}{th}, \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 62.9%

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

   if 3.8e5 < th < 1.0800000000000001e275

   1. Initial program 94.3%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 29.5%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 29.5%

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

   if 1.0800000000000001e275 < th

   1. Initial program 99.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\frac{\sin th}{\sin kx}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\sin kx}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \sin \color{blue}{kx}\right)\right) \]

      3. sin-lowering-sin.f64 47.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{sin.f64}\left(kx\right)\right)\right) \]

   7. Simplified 47.5%

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

Alternative 10: 31.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 3.2 \cdot 10^{-115}:\\ \;\;\;\;\sin th \cdot \left(1 + \frac{kx \cdot -0.5}{\sin ky} \cdot \frac{kx}{\sin ky}\right)\\ \mathbf{elif}\;kx \leq 6 \cdot 10^{-46}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 3.2e-115)
   (* (sin th) (+ 1.0 (* (/ (* kx -0.5) (sin ky)) (/ kx (sin ky)))))
   (if (<= kx 6e-46)
     (* th (/ (sin ky) (hypot kx (sin ky))))
     (* (sin ky) (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.2e-115) {
		tmp = sin(th) * (1.0 + (((kx * -0.5) / sin(ky)) * (kx / sin(ky))));
	} else if (kx <= 6e-46) {
		tmp = th * (sin(ky) / hypot(kx, sin(ky)));
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.2e-115) {
		tmp = Math.sin(th) * (1.0 + (((kx * -0.5) / Math.sin(ky)) * (kx / Math.sin(ky))));
	} else if (kx <= 6e-46) {
		tmp = th * (Math.sin(ky) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 3.2e-115:
		tmp = math.sin(th) * (1.0 + (((kx * -0.5) / math.sin(ky)) * (kx / math.sin(ky))))
	elif kx <= 6e-46:
		tmp = th * (math.sin(ky) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 3.2e-115)
		tmp = Float64(sin(th) * Float64(1.0 + Float64(Float64(Float64(kx * -0.5) / sin(ky)) * Float64(kx / sin(ky)))));
	elseif (kx <= 6e-46)
		tmp = Float64(th * Float64(sin(ky) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 3.2e-115)
		tmp = sin(th) * (1.0 + (((kx * -0.5) / sin(ky)) * (kx / sin(ky))));
	elseif (kx <= 6e-46)
		tmp = th * (sin(ky) / hypot(kx, sin(ky)));
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 3.2e-115], N[(N[Sin[th], $MachinePrecision] * N[(1.0 + N[(N[(N[(kx * -0.5), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(kx / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 6e-46], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 3.2e-115

   1. Initial program 91.0%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}\right)}, \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(\color{blue}{th}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot {kx}^{2}}{{\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot {kx}^{2}\right), \left({\sin ky}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left({kx}^{2}\right)\right), \left({\sin ky}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(kx \cdot kx\right)\right), \left({\sin ky}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(kx, kx\right)\right), \left({\sin ky}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(kx, kx\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      8. sin-lowering-sin.f64 25.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(kx, kx\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\frac{-1}{2} \cdot kx\right) \cdot kx}{{\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\frac{-1}{2} \cdot kx\right) \cdot kx}{\sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot kx}{\sin ky} \cdot \frac{kx}{\sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{2} \cdot kx}{\sin ky}\right), \left(\frac{kx}{\sin ky}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot kx\right), \sin ky\right), \left(\frac{kx}{\sin ky}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(kx \cdot \frac{-1}{2}\right), \sin ky\right), \left(\frac{kx}{\sin ky}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(kx, \frac{-1}{2}\right), \sin ky\right), \left(\frac{kx}{\sin ky}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(kx, \frac{-1}{2}\right), \mathsf{sin.f64}\left(ky\right)\right), \left(\frac{kx}{\sin ky}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(kx, \frac{-1}{2}\right), \mathsf{sin.f64}\left(ky\right)\right), \mathsf{/.f64}\left(kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      10. sin-lowering-sin.f64 29.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(kx, \frac{-1}{2}\right), \mathsf{sin.f64}\left(ky\right)\right), \mathsf{/.f64}\left(kx, \mathsf{sin.f64}\left(ky\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   7. Applied egg-rr 29.1%

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

   if 3.2e-115 < kx < 5.99999999999999975e-46

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.8%

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

   6. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\color{blue}{kx}, \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 99.8%

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

   9. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(kx, \mathsf{sin.f64}\left(ky\right)\right)\right), \color{blue}{th}\right) \]
   10. Step-by-step derivation

   11. Simplified 64.1%

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

   if 5.99999999999999975e-46 < kx

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\frac{\sin th}{\sin kx}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\sin kx}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \sin \color{blue}{kx}\right)\right) \]

      3. sin-lowering-sin.f64 41.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{sin.f64}\left(kx\right)\right)\right) \]

   7. Simplified 41.0%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 3.2 \cdot 10^{-115}:\\ \;\;\;\;\sin th \cdot \left(1 + \frac{kx \cdot -0.5}{\sin ky} \cdot \frac{kx}{\sin ky}\right)\\ \mathbf{elif}\;kx \leq 6 \cdot 10^{-46}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 8.5 \cdot 10^{-47}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.2e-114)
   (sin th)
   (if (<= kx 8.5e-47)
     (* th (/ (sin ky) (hypot kx (sin ky))))
     (* (sin ky) (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.2e-114) {
		tmp = sin(th);
	} else if (kx <= 8.5e-47) {
		tmp = th * (sin(ky) / hypot(kx, sin(ky)));
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.2e-114) {
		tmp = Math.sin(th);
	} else if (kx <= 8.5e-47) {
		tmp = th * (Math.sin(ky) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.2e-114:
		tmp = math.sin(th)
	elif kx <= 8.5e-47:
		tmp = th * (math.sin(ky) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.2e-114)
		tmp = sin(th);
	elseif (kx <= 8.5e-47)
		tmp = Float64(th * Float64(sin(ky) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.2e-114)
		tmp = sin(th);
	elseif (kx <= 8.5e-47)
		tmp = th * (sin(ky) / hypot(kx, sin(ky)));
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.2e-114], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 8.5e-47], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 1.2000000000000001e-114

   1. Initial program 91.0%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 31.1%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 31.1%

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

   if 1.2000000000000001e-114 < kx < 8.4999999999999999e-47

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\sin kx, \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \sin ky\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(kx\right), \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 99.8%

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

   6. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(\color{blue}{kx}, \mathsf{sin.f64}\left(ky\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 99.8%

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

   9. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{hypot.f64}\left(kx, \mathsf{sin.f64}\left(ky\right)\right)\right), \color{blue}{th}\right) \]
   10. Step-by-step derivation

   11. Simplified 64.1%

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

   if 8.4999999999999999e-47 < kx

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\frac{\sin th}{\sin kx}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\sin kx}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \sin \color{blue}{kx}\right)\right) \]

      3. sin-lowering-sin.f64 41.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{sin.f64}\left(kx\right)\right)\right) \]

   7. Simplified 41.0%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 8.5 \cdot 10^{-47}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 31.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.35 \cdot 10^{-113}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.35e-113) (sin th) (* (sin ky) (/ (sin th) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.35e-113) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	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 (kx <= 1.35d-113) then
        tmp = sin(th)
    else
        tmp = sin(ky) * (sin(th) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.35e-113) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.35e-113:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.35e-113)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.35e-113)
		tmp = sin(th);
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.35e-113], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.34999999999999998e-113

   1. Initial program 91.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 30.9%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 30.9%

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

   if 1.34999999999999998e-113 < kx

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \color{blue}{\left(\frac{\sin th}{\sin kx}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\sin kx}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \sin \color{blue}{kx}\right)\right) \]

      3. sin-lowering-sin.f64 41.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{sin.f64}\left(kx\right)\right)\right) \]

   7. Simplified 41.6%

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

Alternative 13: 33.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-160}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx \cdot \left(1 + \left(kx \cdot kx\right) \cdot \left(-0.16666666666666666 + \left(kx \cdot kx\right) \cdot \left(0.008333333333333333 + \left(kx \cdot kx\right) \cdot -0.0001984126984126984\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-160)
   (*
    (sin th)
    (/
     ky
     (*
      kx
      (+
       1.0
       (*
        (* kx kx)
        (+
         -0.16666666666666666
         (*
          (* kx kx)
          (+ 0.008333333333333333 (* (* kx kx) -0.0001984126984126984)))))))))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-160) {
		tmp = sin(th) * (ky / (kx * (1.0 + ((kx * kx) * (-0.16666666666666666 + ((kx * kx) * (0.008333333333333333 + ((kx * kx) * -0.0001984126984126984))))))));
	} 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-160) then
        tmp = sin(th) * (ky / (kx * (1.0d0 + ((kx * kx) * ((-0.16666666666666666d0) + ((kx * kx) * (0.008333333333333333d0 + ((kx * kx) * (-0.0001984126984126984d0)))))))))
    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-160) {
		tmp = Math.sin(th) * (ky / (kx * (1.0 + ((kx * kx) * (-0.16666666666666666 + ((kx * kx) * (0.008333333333333333 + ((kx * kx) * -0.0001984126984126984))))))));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-160:
		tmp = math.sin(th) * (ky / (kx * (1.0 + ((kx * kx) * (-0.16666666666666666 + ((kx * kx) * (0.008333333333333333 + ((kx * kx) * -0.0001984126984126984))))))))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-160)
		tmp = Float64(sin(th) * Float64(ky / Float64(kx * Float64(1.0 + Float64(Float64(kx * kx) * Float64(-0.16666666666666666 + Float64(Float64(kx * kx) * Float64(0.008333333333333333 + Float64(Float64(kx * kx) * -0.0001984126984126984)))))))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-160)
		tmp = sin(th) * (ky / (kx * (1.0 + ((kx * kx) * (-0.16666666666666666 + ((kx * kx) * (0.008333333333333333 + ((kx * kx) * -0.0001984126984126984))))))));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-160], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[(kx * N[(1.0 + N[(N[(kx * kx), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(kx * kx), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(kx * kx), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.9999999999999999e-161

   1. Initial program 90.3%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{ky}{\sin kx}\right)}, \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \sin kx\right), \mathsf{sin.f64}\left(\color{blue}{th}\right)\right) \]

      2. sin-lowering-sin.f64 28.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 28.8%

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

   6. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \color{blue}{\left(kx \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \left({kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({kx}^{2}\right), \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(kx \cdot kx\right), \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \mathsf{+.f64}\left(\left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({kx}^{2}\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(kx \cdot kx\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{-1}{5040} \cdot {kx}^{2}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\frac{-1}{5040}, \left({kx}^{2}\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\frac{-1}{5040}, \left(kx \cdot kx\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      15. *-lowering-*.f64 20.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(kx, kx\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(kx, kx\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   8. Simplified 20.9%

      \[\leadsto \frac{ky}{\color{blue}{kx \cdot \left(1 + \left(kx \cdot kx\right) \cdot \left(\left(kx \cdot kx\right) \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(kx \cdot kx\right)\right) + -0.16666666666666666\right)\right)}} \cdot \sin th \]

   if 9.9999999999999999e-161 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 52.0%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 52.0%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-160}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx \cdot \left(1 + \left(kx \cdot kx\right) \cdot \left(-0.16666666666666666 + \left(kx \cdot kx\right) \cdot \left(0.008333333333333333 + \left(kx \cdot kx\right) \cdot -0.0001984126984126984\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-160}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx \cdot \left(1 + kx \cdot \left(kx \cdot \left(-0.16666666666666666 + \left(kx \cdot kx\right) \cdot 0.008333333333333333\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-160)
   (*
    (sin th)
    (/
     ky
     (*
      kx
      (+
       1.0
       (*
        kx
        (* kx (+ -0.16666666666666666 (* (* kx kx) 0.008333333333333333))))))))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-160) {
		tmp = sin(th) * (ky / (kx * (1.0 + (kx * (kx * (-0.16666666666666666 + ((kx * kx) * 0.008333333333333333)))))));
	} 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-160) then
        tmp = sin(th) * (ky / (kx * (1.0d0 + (kx * (kx * ((-0.16666666666666666d0) + ((kx * kx) * 0.008333333333333333d0)))))))
    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-160) {
		tmp = Math.sin(th) * (ky / (kx * (1.0 + (kx * (kx * (-0.16666666666666666 + ((kx * kx) * 0.008333333333333333)))))));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-160:
		tmp = math.sin(th) * (ky / (kx * (1.0 + (kx * (kx * (-0.16666666666666666 + ((kx * kx) * 0.008333333333333333)))))))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-160)
		tmp = Float64(sin(th) * Float64(ky / Float64(kx * Float64(1.0 + Float64(kx * Float64(kx * Float64(-0.16666666666666666 + Float64(Float64(kx * kx) * 0.008333333333333333))))))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-160)
		tmp = sin(th) * (ky / (kx * (1.0 + (kx * (kx * (-0.16666666666666666 + ((kx * kx) * 0.008333333333333333)))))));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-160], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[(kx * N[(1.0 + N[(kx * N[(kx * N[(-0.16666666666666666 + N[(N[(kx * kx), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.9999999999999999e-161

   1. Initial program 90.3%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{ky}{\sin kx}\right)}, \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \sin kx\right), \mathsf{sin.f64}\left(\color{blue}{th}\right)\right) \]

      2. sin-lowering-sin.f64 28.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 28.8%

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

   6. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \color{blue}{\left(kx \cdot \left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right)\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \left({kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \left(\left(kx \cdot kx\right) \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \left(kx \cdot \left(kx \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(kx, \left(kx \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(kx, \mathsf{*.f64}\left(kx, \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(kx, \mathsf{*.f64}\left(kx, \left(\frac{1}{120} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(kx, \mathsf{*.f64}\left(kx, \left(\frac{1}{120} \cdot {kx}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(kx, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(\left(\frac{1}{120} \cdot {kx}^{2}\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(kx, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{120}, \left({kx}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(kx, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{120}, \left(kx \cdot kx\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      12. *-lowering-*.f64 20.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(kx, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(kx, kx\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   8. Simplified 20.9%

      \[\leadsto \frac{ky}{\color{blue}{kx \cdot \left(1 + kx \cdot \left(kx \cdot \left(0.008333333333333333 \cdot \left(kx \cdot kx\right) + -0.16666666666666666\right)\right)\right)}} \cdot \sin th \]

   if 9.9999999999999999e-161 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 52.0%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 52.0%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-160}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx \cdot \left(1 + kx \cdot \left(kx \cdot \left(-0.16666666666666666 + \left(kx \cdot kx\right) \cdot 0.008333333333333333\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 33.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-160}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx \cdot \left(1 + -0.16666666666666666 \cdot \left(kx \cdot kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-160)
   (* (sin th) (/ ky (* kx (+ 1.0 (* -0.16666666666666666 (* kx kx))))))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-160) {
		tmp = sin(th) * (ky / (kx * (1.0 + (-0.16666666666666666 * (kx * kx)))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 1d-160) then
        tmp = sin(th) * (ky / (kx * (1.0d0 + ((-0.16666666666666666d0) * (kx * kx)))))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 1e-160) {
		tmp = Math.sin(th) * (ky / (kx * (1.0 + (-0.16666666666666666 * (kx * kx)))));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-160:
		tmp = math.sin(th) * (ky / (kx * (1.0 + (-0.16666666666666666 * (kx * kx)))))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-160)
		tmp = Float64(sin(th) * Float64(ky / Float64(kx * Float64(1.0 + Float64(-0.16666666666666666 * Float64(kx * kx))))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-160)
		tmp = sin(th) * (ky / (kx * (1.0 + (-0.16666666666666666 * (kx * kx)))));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-160], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[(kx * N[(1.0 + N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.9999999999999999e-161

   1. Initial program 90.3%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{ky}{\sin kx}\right)}, \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \sin kx\right), \mathsf{sin.f64}\left(\color{blue}{th}\right)\right) \]

      2. sin-lowering-sin.f64 28.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 28.8%

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

   6. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \color{blue}{\left(kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)\right)}\right), \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {kx}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left({kx}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(kx \cdot kx\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

      5. *-lowering-*.f64 20.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{*.f64}\left(kx, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(kx, kx\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   8. Simplified 20.9%

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

   if 9.9999999999999999e-161 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 52.0%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 52.0%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-160}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx \cdot \left(1 + -0.16666666666666666 \cdot \left(kx \cdot kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.4 \cdot 10^{-109}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.4e-109) (sin th) (* ky (/ (sin th) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.4e-109) {
		tmp = sin(th);
	} else {
		tmp = ky * (sin(th) / sin(kx));
	}
	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 (kx <= 1.4d-109) then
        tmp = sin(th)
    else
        tmp = ky * (sin(th) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.4e-109) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.4e-109:
		tmp = math.sin(th)
	else:
		tmp = ky * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.4e-109)
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.4e-109)
		tmp = sin(th);
	else
		tmp = ky * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.4e-109], N[Sin[th], $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.39999999999999989e-109

   1. Initial program 91.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 30.8%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 30.8%

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

   if 1.39999999999999989e-109 < kx

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{ky}{\sin kx}\right)}, \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \sin kx\right), \mathsf{sin.f64}\left(\color{blue}{th}\right)\right) \]

      2. sin-lowering-sin.f64 38.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 38.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(ky, \color{blue}{\left(\frac{\sin th}{\sin kx}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(ky, \mathsf{/.f64}\left(\sin th, \color{blue}{\sin kx}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(ky, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \sin \color{blue}{kx}\right)\right) \]

      6. sin-lowering-sin.f64 38.7%

         \[\leadsto \mathsf{*.f64}\left(ky, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{sin.f64}\left(kx\right)\right)\right) \]

   7. Applied egg-rr 38.7%

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

Alternative 17: 26.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2.3 \cdot 10^{-160}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 2.3e-160) (* (sin th) (/ ky kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.3e-160) {
		tmp = sin(th) * (ky / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 2.3d-160) then
        tmp = sin(th) * (ky / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.3e-160) {
		tmp = Math.sin(th) * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 2.3e-160:
		tmp = math.sin(th) * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 2.3e-160)
		tmp = Float64(sin(th) * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 2.3e-160)
		tmp = sin(th) * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 2.3e-160], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.29999999999999985e-160

   1. Initial program 90.0%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{ky}{\sin kx}\right)}, \mathsf{sin.f64}\left(th\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \sin kx\right), \mathsf{sin.f64}\left(\color{blue}{th}\right)\right) \]

      2. sin-lowering-sin.f64 29.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, \mathsf{sin.f64}\left(kx\right)\right), \mathsf{sin.f64}\left(th\right)\right) \]

   5. Simplified 29.5%

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

   6. Taylor expanded in kx around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{ky}{kx}\right)}, \mathsf{sin.f64}\left(th\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 21.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(ky, kx\right), \mathsf{sin.f64}\left(\color{blue}{th}\right)\right) \]

   8. Simplified 21.4%

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

   if 2.29999999999999985e-160 < ky

   1. Initial program 99.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 37.1%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 37.1%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 2.3 \cdot 10^{-160}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 24.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 3.1 \cdot 10^{+46}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \left(-0.16666666666666666 \cdot \left(th \cdot th\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 3.1e+46) (sin th) (* th (* -0.16666666666666666 (* th th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.1e+46) {
		tmp = sin(th);
	} else {
		tmp = th * (-0.16666666666666666 * (th * 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 (kx <= 3.1d+46) then
        tmp = sin(th)
    else
        tmp = th * ((-0.16666666666666666d0) * (th * th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.1e+46) {
		tmp = Math.sin(th);
	} else {
		tmp = th * (-0.16666666666666666 * (th * th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 3.1e+46:
		tmp = math.sin(th)
	else:
		tmp = th * (-0.16666666666666666 * (th * th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 3.1e+46)
		tmp = sin(th);
	else
		tmp = Float64(th * Float64(-0.16666666666666666 * Float64(th * th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 3.1e+46)
		tmp = sin(th);
	else
		tmp = th * (-0.16666666666666666 * (th * th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 3.1e+46], N[Sin[th], $MachinePrecision], N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 3.09999999999999975e46

   1. Initial program 92.0%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 29.0%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 29.0%

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

   if 3.09999999999999975e46 < kx

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 7.7%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 7.7%

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

   8. Taylor expanded in th around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(th, \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({th}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(th \cdot \color{blue}{th}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 3.8%

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, \color{blue}{th}\right)\right)\right)\right) \]

   10. Simplified 3.8%

       \[\leadsto \color{blue}{th \cdot \left(1 + -0.16666666666666666 \cdot \left(th \cdot th\right)\right)} \]

   11. Taylor expanded in th around inf

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

       1. *-commutative N/A

          \[\leadsto {th}^{3} \cdot \color{blue}{\frac{-1}{6}} \]

       2. cube-mult N/A

          \[\leadsto \left(th \cdot \left(th \cdot th\right)\right) \cdot \frac{-1}{6} \]

       3. unpow2 N/A

          \[\leadsto \left(th \cdot {th}^{2}\right) \cdot \frac{-1}{6} \]

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(th, \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({th}^{2}\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(\frac{-1}{6}, \left(th \cdot \color{blue}{th}\right)\right)\right) \]

       9. *-lowering-*.f64 15.9%

          \[\leadsto \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, \color{blue}{th}\right)\right)\right) \]

   13. Simplified 15.9%

       \[\leadsto \color{blue}{th \cdot \left(-0.16666666666666666 \cdot \left(th \cdot th\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 15.4% accurate, 59.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.25 \cdot 10^{-53}:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \left(-0.16666666666666666 \cdot \left(th \cdot th\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.25e-53) th (* th (* -0.16666666666666666 (* th th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.25e-53) {
		tmp = th;
	} else {
		tmp = th * (-0.16666666666666666 * (th * 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 (kx <= 1.25d-53) then
        tmp = th
    else
        tmp = th * ((-0.16666666666666666d0) * (th * th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.25e-53) {
		tmp = th;
	} else {
		tmp = th * (-0.16666666666666666 * (th * th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.25e-53:
		tmp = th
	else:
		tmp = th * (-0.16666666666666666 * (th * th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.25e-53)
		tmp = th;
	else
		tmp = Float64(th * Float64(-0.16666666666666666 * Float64(th * th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.25e-53)
		tmp = th;
	else
		tmp = th * (-0.16666666666666666 * (th * th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.25e-53], th, N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.25e-53

   1. Initial program 91.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 30.4%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 30.4%

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

   8. Taylor expanded in th around 0

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

   10. Simplified 15.1%

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

   if 1.25e-53 < kx

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

      10. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

      13. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 8.4%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   7. Simplified 8.4%

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

   8. Taylor expanded in th around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(th, \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({th}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(th \cdot \color{blue}{th}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 4.3%

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, \color{blue}{th}\right)\right)\right)\right) \]

   10. Simplified 4.3%

       \[\leadsto \color{blue}{th \cdot \left(1 + -0.16666666666666666 \cdot \left(th \cdot th\right)\right)} \]

   11. Taylor expanded in th around inf

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

       1. *-commutative N/A

          \[\leadsto {th}^{3} \cdot \color{blue}{\frac{-1}{6}} \]

       2. cube-mult N/A

          \[\leadsto \left(th \cdot \left(th \cdot th\right)\right) \cdot \frac{-1}{6} \]

       3. unpow2 N/A

          \[\leadsto \left(th \cdot {th}^{2}\right) \cdot \frac{-1}{6} \]

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(th, \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({th}^{2}\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(\frac{-1}{6}, \left(th \cdot \color{blue}{th}\right)\right)\right) \]

       9. *-lowering-*.f64 15.8%

          \[\leadsto \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, \color{blue}{th}\right)\right)\right) \]

   13. Simplified 15.8%

       \[\leadsto \color{blue}{th \cdot \left(-0.16666666666666666 \cdot \left(th \cdot th\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 13.3% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(kx, ky, th):
	return th

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.8%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

   10. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

   13. sin-lowering-sin.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

3. Simplified 99.6%

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

5. Taylor expanded in kx around 0

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

   1. sin-lowering-sin.f64 23.8%

      \[\leadsto \mathsf{sin.f64}\left(th\right) \]

7. Simplified 23.8%

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

8. Taylor expanded in th around 0

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

10. Simplified 11.9%

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

Reproduce

herbie shell --seed 2024145 
(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)))