Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.7%

Time: 27.5s

Alternatives: 19

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: 93.8% 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 92.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ky \cdot \left(1 + \left(ky \cdot ky\right) \cdot -0.16666666666666666\right)\\ \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;0 - \sin th\\ \mathbf{elif}\;\sin ky \leq 0.03:\\ \;\;\;\;\frac{t\_1}{\frac{\mathsf{hypot}\left(\sin kx, t\_1\right)}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 0.385:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* ky (+ 1.0 (* (* ky ky) -0.16666666666666666)))))
   (if (<= (sin ky) -0.02)
     (- 0.0 (sin th))
     (if (<= (sin ky) 0.03)
       (/ t_1 (/ (hypot (sin kx) t_1) (sin th)))
       (if (<= (sin ky) 0.385)
         (sin th)
         (* (sin ky) (/ th (hypot (sin ky) (sin kx)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = ky * (1.0 + ((ky * ky) * -0.16666666666666666));
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = 0.0 - sin(th);
	} else if (sin(ky) <= 0.03) {
		tmp = t_1 / (hypot(sin(kx), t_1) / sin(th));
	} else if (sin(ky) <= 0.385) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = ky * (1.0 + ((ky * ky) * -0.16666666666666666));
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = 0.0 - Math.sin(th);
	} else if (Math.sin(ky) <= 0.03) {
		tmp = t_1 / (Math.hypot(Math.sin(kx), t_1) / Math.sin(th));
	} else if (Math.sin(ky) <= 0.385) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = ky * (1.0 + ((ky * ky) * -0.16666666666666666))
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = 0.0 - math.sin(th)
	elif math.sin(ky) <= 0.03:
		tmp = t_1 / (math.hypot(math.sin(kx), t_1) / math.sin(th))
	elif math.sin(ky) <= 0.385:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(ky * Float64(1.0 + Float64(Float64(ky * ky) * -0.16666666666666666)))
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(0.0 - sin(th));
	elseif (sin(ky) <= 0.03)
		tmp = Float64(t_1 / Float64(hypot(sin(kx), t_1) / sin(th)));
	elseif (sin(ky) <= 0.385)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = ky * (1.0 + ((ky * ky) * -0.16666666666666666));
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = 0.0 - sin(th);
	elseif (sin(ky) <= 0.03)
		tmp = t_1 / (hypot(sin(kx), t_1) / sin(th));
	elseif (sin(ky) <= 0.385)
		tmp = sin(th);
	else
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[(1.0 + N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(0.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.03], N[(t$95$1 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.385], N[Sin[th], $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 4 regimes

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

   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 ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 12.4%

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

   10. Simplified 12.4%

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

   11. Taylor expanded in ky around inf

       \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\sin th} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\sin th}\right) \]

       4. sin-lowering-sin.f64 61.3%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(th\right)\right) \]

   13. Simplified 61.3%

       \[\leadsto \color{blue}{0 - \sin th} \]

   if -0.0200000000000000004 < (sin.f64 ky) < 0.029999999999999999

   1. Initial program 86.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.6%

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

   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

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 98.2%

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

   10. Simplified 98.2%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. Applied egg-rr 98.2%

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

   if 0.029999999999999999 < (sin.f64 ky) < 0.38500000000000001

   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 99.5%

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

   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 74.1%

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

   7. Simplified 74.1%

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

   if 0.38500000000000001 < (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 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 45.2%

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

Alternative 3: 79.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{{\left(0.5 + -0.5 \cdot \cos \left(kx \cdot 2\right)\right)}^{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 5e-14)
   (* (sin th) (/ (sin ky) (hypot kx (sin ky))))
   (* (sin th) (/ (sin ky) (pow (+ 0.5 (* -0.5 (cos (* kx 2.0)))) 0.5)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 5e-14) {
		tmp = sin(th) * (sin(ky) / hypot(kx, sin(ky)));
	} else {
		tmp = sin(th) * (sin(ky) / pow((0.5 + (-0.5 * cos((kx * 2.0)))), 0.5));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.pow(Math.sin(kx), 2.0) <= 5e-14) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.pow((0.5 + (-0.5 * Math.cos((kx * 2.0)))), 0.5));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.pow(math.sin(kx), 2.0) <= 5e-14:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.pow((0.5 + (-0.5 * math.cos((kx * 2.0)))), 0.5))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 5e-14)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / (Float64(0.5 + Float64(-0.5 * cos(Float64(kx * 2.0)))) ^ 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(kx) ^ 2.0) <= 5e-14)
		tmp = sin(th) * (sin(ky) / hypot(kx, sin(ky)));
	else
		tmp = sin(th) * (sin(ky) / ((0.5 + (-0.5 * cos((kx * 2.0)))) ^ 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 5e-14], 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[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Power[N[(0.5 + N[(-0.5 * N[Cos[N[(kx * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 5.0000000000000002e-14

   1. Initial program 85.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.9%

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

   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 99.9%

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

   if 5.0000000000000002e-14 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 40.0%

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

   5. Simplified 40.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

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

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

      6. sqr-sin-a N/A

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

      11. count-2 N/A

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

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

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

      13. count-2 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 64.6%

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

   7. Applied egg-rr 64.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{{\left(0.5 + -0.5 \cdot \cos \left(kx \cdot 2\right)\right)}^{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ky \cdot \left(1 + \left(ky \cdot ky\right) \cdot -0.16666666666666666\right)\\ \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;0 - \sin th\\ \mathbf{elif}\;\sin ky \leq 0.03:\\ \;\;\;\;\frac{t\_1}{\frac{\mathsf{hypot}\left(\sin kx, t\_1\right)}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* ky (+ 1.0 (* (* ky ky) -0.16666666666666666)))))
   (if (<= (sin ky) -0.02)
     (- 0.0 (sin th))
     (if (<= (sin ky) 0.03)
       (/ t_1 (/ (hypot (sin kx) t_1) (sin th)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = ky * (1.0 + ((ky * ky) * -0.16666666666666666));
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = 0.0 - sin(th);
	} else if (sin(ky) <= 0.03) {
		tmp = t_1 / (hypot(sin(kx), t_1) / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = ky * (1.0 + ((ky * ky) * -0.16666666666666666));
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = 0.0 - Math.sin(th);
	} else if (Math.sin(ky) <= 0.03) {
		tmp = t_1 / (Math.hypot(Math.sin(kx), t_1) / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = ky * (1.0 + ((ky * ky) * -0.16666666666666666))
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = 0.0 - math.sin(th)
	elif math.sin(ky) <= 0.03:
		tmp = t_1 / (math.hypot(math.sin(kx), t_1) / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(ky * Float64(1.0 + Float64(Float64(ky * ky) * -0.16666666666666666)))
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(0.0 - sin(th));
	elseif (sin(ky) <= 0.03)
		tmp = Float64(t_1 / Float64(hypot(sin(kx), t_1) / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = ky * (1.0 + ((ky * ky) * -0.16666666666666666));
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = 0.0 - sin(th);
	elseif (sin(ky) <= 0.03)
		tmp = t_1 / (hypot(sin(kx), t_1) / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[(1.0 + N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(0.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.03], N[(t$95$1 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   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 ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 12.4%

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

   10. Simplified 12.4%

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

   11. Taylor expanded in ky around inf

       \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\sin th} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\sin th}\right) \]

       4. sin-lowering-sin.f64 61.3%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(th\right)\right) \]

   13. Simplified 61.3%

       \[\leadsto \color{blue}{0 - \sin th} \]

   if -0.0200000000000000004 < (sin.f64 ky) < 0.029999999999999999

   1. Initial program 86.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.6%

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

   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

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 98.2%

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

   10. Simplified 98.2%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. Applied egg-rr 98.2%

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

   if 0.029999999999999999 < (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 99.5%

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

   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 59.7%

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

   7. Simplified 59.7%

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

Alternative 5: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ky \cdot \left(1 + \left(ky \cdot ky\right) \cdot -0.16666666666666666\right)\\ \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;0 - \sin th\\ \mathbf{elif}\;\sin ky \leq 0.03:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\mathsf{hypot}\left(t\_1, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* ky (+ 1.0 (* (* ky ky) -0.16666666666666666)))))
   (if (<= (sin ky) -0.02)
     (- 0.0 (sin th))
     (if (<= (sin ky) 0.03)
       (* t_1 (/ (sin th) (hypot t_1 (sin kx))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = ky * (1.0 + ((ky * ky) * -0.16666666666666666));
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = 0.0 - sin(th);
	} else if (sin(ky) <= 0.03) {
		tmp = t_1 * (sin(th) / hypot(t_1, sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = ky * (1.0 + ((ky * ky) * -0.16666666666666666));
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = 0.0 - Math.sin(th);
	} else if (Math.sin(ky) <= 0.03) {
		tmp = t_1 * (Math.sin(th) / Math.hypot(t_1, Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = ky * (1.0 + ((ky * ky) * -0.16666666666666666))
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = 0.0 - math.sin(th)
	elif math.sin(ky) <= 0.03:
		tmp = t_1 * (math.sin(th) / math.hypot(t_1, math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(ky * Float64(1.0 + Float64(Float64(ky * ky) * -0.16666666666666666)))
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(0.0 - sin(th));
	elseif (sin(ky) <= 0.03)
		tmp = Float64(t_1 * Float64(sin(th) / hypot(t_1, sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = ky * (1.0 + ((ky * ky) * -0.16666666666666666));
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = 0.0 - sin(th);
	elseif (sin(ky) <= 0.03)
		tmp = t_1 * (sin(th) / hypot(t_1, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[(1.0 + N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(0.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.03], N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[t$95$1 ^ 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) < -0.0200000000000000004

   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 ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 12.4%

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

   10. Simplified 12.4%

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

   11. Taylor expanded in ky around inf

       \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\sin th} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\sin th}\right) \]

       4. sin-lowering-sin.f64 61.3%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(th\right)\right) \]

   13. Simplified 61.3%

       \[\leadsto \color{blue}{0 - \sin th} \]

   if -0.0200000000000000004 < (sin.f64 ky) < 0.029999999999999999

   1. Initial program 86.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.6%

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

   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

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 98.2%

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

   10. Simplified 98.2%

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

   if 0.029999999999999999 < (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 99.5%

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

   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 59.7%

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

   7. Simplified 59.7%

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

Alternative 6: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;0 - \sin th\\ \mathbf{elif}\;\sin ky \leq 10^{-12}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky \cdot \left(1 + \left(ky \cdot ky\right) \cdot -0.16666666666666666\right), \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (- 0.0 (sin th))
   (if (<= (sin ky) 1e-12)
     (*
      ky
      (/
       (sin th)
       (hypot (* ky (+ 1.0 (* (* ky ky) -0.16666666666666666))) (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = 0.0 - sin(th);
	} else if (sin(ky) <= 1e-12) {
		tmp = ky * (sin(th) / hypot((ky * (1.0 + ((ky * ky) * -0.16666666666666666))), 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) <= -0.02) {
		tmp = 0.0 - Math.sin(th);
	} else if (Math.sin(ky) <= 1e-12) {
		tmp = ky * (Math.sin(th) / Math.hypot((ky * (1.0 + ((ky * ky) * -0.16666666666666666))), Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = 0.0 - math.sin(th)
	elif math.sin(ky) <= 1e-12:
		tmp = ky * (math.sin(th) / math.hypot((ky * (1.0 + ((ky * ky) * -0.16666666666666666))), math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(0.0 - sin(th));
	elseif (sin(ky) <= 1e-12)
		tmp = Float64(ky * Float64(sin(th) / hypot(Float64(ky * Float64(1.0 + Float64(Float64(ky * ky) * -0.16666666666666666))), sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = 0.0 - sin(th);
	elseif (sin(ky) <= 1e-12)
		tmp = ky * (sin(th) / hypot((ky * (1.0 + ((ky * ky) * -0.16666666666666666))), sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(0.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-12], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(ky * N[(1.0 + N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 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) < -0.0200000000000000004

   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 ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 12.4%

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

   10. Simplified 12.4%

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

   11. Taylor expanded in ky around inf

       \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\sin th} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\sin th}\right) \]

       4. sin-lowering-sin.f64 61.3%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(th\right)\right) \]

   13. Simplified 61.3%

       \[\leadsto \color{blue}{0 - \sin th} \]

   if -0.0200000000000000004 < (sin.f64 ky) < 9.9999999999999998e-13

   1. Initial program 86.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in ky around 0

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

   10. Simplified 98.9%

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

   if 9.9999999999999998e-13 < (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 99.5%

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

   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 60.9%

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

   7. Simplified 60.9%

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

Alternative 7: 54.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;0 - \sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (- 0.0 (sin th))
   (if (<= (sin ky) 2e-50) (/ (sin th) (/ (sin kx) (sin ky))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = 0.0 - sin(th);
	} else if (sin(ky) <= 2e-50) {
		tmp = sin(th) / (sin(kx) / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.02d0)) then
        tmp = 0.0d0 - sin(th)
    else if (sin(ky) <= 2d-50) then
        tmp = sin(th) / (sin(kx) / sin(ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = 0.0 - Math.sin(th);
	} else if (Math.sin(ky) <= 2e-50) {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = 0.0 - math.sin(th)
	elif math.sin(ky) <= 2e-50:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(0.0 - sin(th));
	elseif (sin(ky) <= 2e-50)
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = 0.0 - sin(th);
	elseif (sin(ky) <= 2e-50)
		tmp = sin(th) / (sin(kx) / sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(0.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-50], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   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 ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 12.4%

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

   10. Simplified 12.4%

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

   11. Taylor expanded in ky around inf

       \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\sin th} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\sin th}\right) \]

       4. sin-lowering-sin.f64 61.3%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(th\right)\right) \]

   13. Simplified 61.3%

       \[\leadsto \color{blue}{0 - \sin th} \]

   if -0.0200000000000000004 < (sin.f64 ky) < 2.00000000000000002e-50

   1. Initial program 85.8%

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

   3. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 48.6%

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

   5. Simplified 48.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      8. sin-lowering-sin.f64 48.7%

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

   7. Applied egg-rr 48.7%

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

   if 2.00000000000000002e-50 < (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 99.5%

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

   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 62.0%

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

   7. Simplified 62.0%

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

Alternative 8: 54.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;0 - \sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (- 0.0 (sin th))
   (if (<= (sin ky) 2e-50) (* (sin ky) (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = 0.0 - sin(th);
	} else if (sin(ky) <= 2e-50) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.02d0)) then
        tmp = 0.0d0 - sin(th)
    else if (sin(ky) <= 2d-50) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = 0.0 - Math.sin(th);
	} else if (Math.sin(ky) <= 2e-50) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = 0.0 - math.sin(th)
	elif math.sin(ky) <= 2e-50:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(0.0 - sin(th));
	elseif (sin(ky) <= 2e-50)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = 0.0 - sin(th);
	elseif (sin(ky) <= 2e-50)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(0.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-50], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   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 ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 12.4%

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

   10. Simplified 12.4%

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

   11. Taylor expanded in ky around inf

       \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\sin th} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\sin th}\right) \]

       4. sin-lowering-sin.f64 61.3%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(th\right)\right) \]

   13. Simplified 61.3%

       \[\leadsto \color{blue}{0 - \sin th} \]

   if -0.0200000000000000004 < (sin.f64 ky) < 2.00000000000000002e-50

   1. Initial program 85.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 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 48.6%

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

   7. Simplified 48.6%

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

   if 2.00000000000000002e-50 < (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 99.5%

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

   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 62.0%

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

   7. Simplified 62.0%

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

Alternative 9: 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 92.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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 10: 54.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;0 - \sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\left(ky \cdot \left(1 + \left(ky \cdot ky\right) \cdot -0.16666666666666666\right)\right) \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (- 0.0 (sin th))
   (if (<= (sin ky) 2e-50)
     (*
      (* ky (+ 1.0 (* (* ky ky) -0.16666666666666666)))
      (/ (sin th) (sin kx)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = 0.0 - sin(th);
	} else if (sin(ky) <= 2e-50) {
		tmp = (ky * (1.0 + ((ky * ky) * -0.16666666666666666))) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.02d0)) then
        tmp = 0.0d0 - sin(th)
    else if (sin(ky) <= 2d-50) then
        tmp = (ky * (1.0d0 + ((ky * ky) * (-0.16666666666666666d0)))) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = 0.0 - Math.sin(th);
	} else if (Math.sin(ky) <= 2e-50) {
		tmp = (ky * (1.0 + ((ky * ky) * -0.16666666666666666))) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = 0.0 - math.sin(th)
	elif math.sin(ky) <= 2e-50:
		tmp = (ky * (1.0 + ((ky * ky) * -0.16666666666666666))) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(0.0 - sin(th));
	elseif (sin(ky) <= 2e-50)
		tmp = Float64(Float64(ky * Float64(1.0 + Float64(Float64(ky * ky) * -0.16666666666666666))) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = 0.0 - sin(th);
	elseif (sin(ky) <= 2e-50)
		tmp = (ky * (1.0 + ((ky * ky) * -0.16666666666666666))) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(0.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-50], N[(N[(ky * N[(1.0 + N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   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 ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 12.4%

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

   10. Simplified 12.4%

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

   11. Taylor expanded in ky around inf

       \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\sin th} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\sin th}\right) \]

       4. sin-lowering-sin.f64 61.3%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(th\right)\right) \]

   13. Simplified 61.3%

       \[\leadsto \color{blue}{0 - \sin th} \]

   if -0.0200000000000000004 < (sin.f64 ky) < 2.00000000000000002e-50

   1. Initial program 85.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 98.8%

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

   10. Simplified 98.8%

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

   11. Taylor expanded in ky around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 48.6%

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

   13. Simplified 48.6%

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

   if 2.00000000000000002e-50 < (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 99.5%

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

   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 62.0%

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

   7. Simplified 62.0%

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

Alternative 11: 54.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;0 - \sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (- 0.0 (sin th))
   (if (<= (sin ky) 2e-50) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = 0.0 - sin(th);
	} else if (sin(ky) <= 2e-50) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.02d0)) then
        tmp = 0.0d0 - sin(th)
    else if (sin(ky) <= 2d-50) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = 0.0 - Math.sin(th);
	} else if (Math.sin(ky) <= 2e-50) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = 0.0 - math.sin(th)
	elif math.sin(ky) <= 2e-50:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(0.0 - sin(th));
	elseif (sin(ky) <= 2e-50)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = 0.0 - sin(th);
	elseif (sin(ky) <= 2e-50)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(0.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-50], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   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 ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 12.4%

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

   10. Simplified 12.4%

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

   11. Taylor expanded in ky around inf

       \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\sin th} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\sin th}\right) \]

       4. sin-lowering-sin.f64 61.3%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(th\right)\right) \]

   13. Simplified 61.3%

       \[\leadsto \color{blue}{0 - \sin th} \]

   if -0.0200000000000000004 < (sin.f64 ky) < 2.00000000000000002e-50

   1. Initial program 85.8%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{ky}{\sin kx}\right)}, \mathsf{sin.f64}\left(th\right)\right) \]
   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 48.6%

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

   5. Simplified 48.6%

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

   if 2.00000000000000002e-50 < (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 99.5%

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

   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 62.0%

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

   7. Simplified 62.0%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;0 - \sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;0 - \sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (- 0.0 (sin th))
   (if (<= (sin ky) 2e-74) (* (sin ky) (/ (sin th) kx)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = 0.0 - sin(th);
	} else if (sin(ky) <= 2e-74) {
		tmp = sin(ky) * (sin(th) / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.02d0)) then
        tmp = 0.0d0 - sin(th)
    else if (sin(ky) <= 2d-74) then
        tmp = sin(ky) * (sin(th) / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = 0.0 - Math.sin(th);
	} else if (Math.sin(ky) <= 2e-74) {
		tmp = Math.sin(ky) * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = 0.0 - math.sin(th)
	elif math.sin(ky) <= 2e-74:
		tmp = math.sin(ky) * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(0.0 - sin(th));
	elseif (sin(ky) <= 2e-74)
		tmp = Float64(sin(ky) * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = 0.0 - sin(th);
	elseif (sin(ky) <= 2e-74)
		tmp = sin(ky) * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(0.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-74], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   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 ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 12.4%

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

   10. Simplified 12.4%

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

   11. Taylor expanded in ky around inf

       \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\sin th} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\sin th}\right) \]

       4. sin-lowering-sin.f64 61.3%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(th\right)\right) \]

   13. Simplified 61.3%

       \[\leadsto \color{blue}{0 - \sin th} \]

   if -0.0200000000000000004 < (sin.f64 ky) < 1.99999999999999992e-74

   1. Initial program 85.3%

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

   3. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 48.2%

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

   5. Simplified 48.2%

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

   6. Taylor expanded in kx around 0

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sin-lowering-sin.f64 28.7%

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

   8. Simplified 28.7%

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

   if 1.99999999999999992e-74 < (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 59.7%

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

   7. Simplified 59.7%

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

Alternative 13: 46.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;0 - \sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-74}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (- 0.0 (sin th))
   (if (<= (sin ky) 2e-74) (* th (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = 0.0 - sin(th);
	} else if (sin(ky) <= 2e-74) {
		tmp = th * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.02d0)) then
        tmp = 0.0d0 - sin(th)
    else if (sin(ky) <= 2d-74) then
        tmp = th * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = 0.0 - Math.sin(th);
	} else if (Math.sin(ky) <= 2e-74) {
		tmp = th * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = 0.0 - math.sin(th)
	elif math.sin(ky) <= 2e-74:
		tmp = th * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(0.0 - sin(th));
	elseif (sin(ky) <= 2e-74)
		tmp = Float64(th * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = 0.0 - sin(th);
	elseif (sin(ky) <= 2e-74)
		tmp = th * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(0.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-74], N[(th * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   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 ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 12.4%

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

   10. Simplified 12.4%

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

   11. Taylor expanded in ky around inf

       \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\sin th} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\sin th}\right) \]

       4. sin-lowering-sin.f64 61.3%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(th\right)\right) \]

   13. Simplified 61.3%

       \[\leadsto \color{blue}{0 - \sin th} \]

   if -0.0200000000000000004 < (sin.f64 ky) < 1.99999999999999992e-74

   1. Initial program 85.3%

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

   3. Taylor expanded in th around 0

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

   5. Simplified 41.6%

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

   6. Taylor expanded in ky around 0

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

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

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

      2. sin-lowering-sin.f64 25.3%

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

   8. Simplified 25.3%

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

   if 1.99999999999999992e-74 < (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 59.7%

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

   7. Simplified 59.7%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;0 - \sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-74}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 6.6 \cdot 10^{-59}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 2.35 \cdot 10^{-7}:\\ \;\;\;\;0 - \sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 6.6e-59)
   (sin th)
   (if (<= kx 2.35e-7) (- 0.0 (sin th)) (* ky (/ th (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.6e-59) {
		tmp = sin(th);
	} else if (kx <= 2.35e-7) {
		tmp = 0.0 - sin(th);
	} else {
		tmp = ky * (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 <= 6.6d-59) then
        tmp = sin(th)
    else if (kx <= 2.35d-7) then
        tmp = 0.0d0 - sin(th)
    else
        tmp = ky * (th / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.6e-59) {
		tmp = Math.sin(th);
	} else if (kx <= 2.35e-7) {
		tmp = 0.0 - Math.sin(th);
	} else {
		tmp = ky * (th / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 6.6e-59:
		tmp = math.sin(th)
	elif kx <= 2.35e-7:
		tmp = 0.0 - math.sin(th)
	else:
		tmp = ky * (th / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 6.6e-59)
		tmp = sin(th);
	elseif (kx <= 2.35e-7)
		tmp = Float64(0.0 - sin(th));
	else
		tmp = Float64(ky * Float64(th / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 6.6e-59)
		tmp = sin(th);
	elseif (kx <= 2.35e-7)
		tmp = 0.0 - sin(th);
	else
		tmp = ky * (th / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 6.6e-59], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 2.35e-7], N[(0.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 6.59999999999999964e-59

   1. Initial program 89.9%

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

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

      1. sin-lowering-sin.f64 29.8%

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

   7. Simplified 29.8%

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

   if 6.59999999999999964e-59 < kx < 2.35e-7

   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 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 34.3%

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

   7. Simplified 34.3%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 49.8%

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

   10. Simplified 49.8%

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

   11. Taylor expanded in ky around inf

       \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\sin th} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\sin th}\right) \]

       4. sin-lowering-sin.f64 45.0%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(th\right)\right) \]

   13. Simplified 45.0%

       \[\leadsto \color{blue}{0 - \sin th} \]

   if 2.35e-7 < kx

   1. Initial program 99.3%

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

   3. Taylor expanded in th around 0

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

   5. Simplified 44.5%

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

   6. Taylor expanded in ky around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 17.5%

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

   8. Simplified 17.5%

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

Alternative 15: 23.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.46 \cdot 10^{-233}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{elif}\;ky \leq 3.65 \cdot 10^{+49}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;0 - \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.46e-233)
   (/ th (/ kx ky))
   (if (<= ky 3.65e+49) (sin th) (- 0.0 (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.46e-233) {
		tmp = th / (kx / ky);
	} else if (ky <= 3.65e+49) {
		tmp = sin(th);
	} else {
		tmp = 0.0 - 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 <= 1.46d-233) then
        tmp = th / (kx / ky)
    else if (ky <= 3.65d+49) then
        tmp = sin(th)
    else
        tmp = 0.0d0 - sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.46e-233) {
		tmp = th / (kx / ky);
	} else if (ky <= 3.65e+49) {
		tmp = Math.sin(th);
	} else {
		tmp = 0.0 - Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.46e-233:
		tmp = th / (kx / ky)
	elif ky <= 3.65e+49:
		tmp = math.sin(th)
	else:
		tmp = 0.0 - math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.46e-233)
		tmp = Float64(th / Float64(kx / ky));
	elseif (ky <= 3.65e+49)
		tmp = sin(th);
	else
		tmp = Float64(0.0 - sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.46e-233)
		tmp = th / (kx / ky);
	elseif (ky <= 3.65e+49)
		tmp = sin(th);
	else
		tmp = 0.0 - sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.46e-233], N[(th / N[(kx / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 3.65e+49], N[Sin[th], $MachinePrecision], N[(0.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 1.46000000000000011e-233

   1. Initial program 91.9%

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

   3. Taylor expanded in th around 0

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

   5. Simplified 45.7%

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

   6. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 31.2%

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

   8. Simplified 31.2%

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

   9. Taylor expanded in ky around 0

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

       1. /-lowering-/.f64 13.4%

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

   11. Simplified 13.4%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

       5. /-lowering-/.f64 13.5%

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

   13. Applied egg-rr 13.5%

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

   if 1.46000000000000011e-233 < ky < 3.65000000000000007e49

   1. Initial program 90.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.5%

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

   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 33.4%

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

   7. Simplified 33.4%

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

   if 3.65000000000000007e49 < 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 99.5%

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

   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

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 4.7%

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

   7. Simplified 4.7%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 6.3%

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

   10. Simplified 6.3%

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

   11. Taylor expanded in ky around inf

       \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\sin th} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\sin th}\right) \]

       4. sin-lowering-sin.f64 32.6%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(th\right)\right) \]

   13. Simplified 32.6%

       \[\leadsto \color{blue}{0 - \sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 25.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 6.6 \cdot 10^{-6}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot th}{kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 6.6e-6) (sin th) (/ (* ky th) kx)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.6e-6) {
		tmp = sin(th);
	} else {
		tmp = (ky * th) / 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 <= 6.6d-6) then
        tmp = sin(th)
    else
        tmp = (ky * th) / kx
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.6e-6) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky * th) / kx;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 6.6e-6:
		tmp = math.sin(th)
	else:
		tmp = (ky * th) / kx
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 6.6e-6)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky * th) / kx);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 6.6e-6)
		tmp = sin(th);
	else
		tmp = (ky * th) / kx;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 6.6e-6], N[Sin[th], $MachinePrecision], N[(N[(ky * th), $MachinePrecision] / kx), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 6.60000000000000034e-6

   1. Initial program 90.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 30.6%

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

   7. Simplified 30.6%

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

   if 6.60000000000000034e-6 < kx

   1. Initial program 99.3%

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

   3. Taylor expanded in th around 0

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

   5. Simplified 44.5%

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

   6. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 15.2%

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

   8. Simplified 15.2%

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

   9. Taylor expanded in ky around 0

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

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

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

       2. *-lowering-*.f64 15.3%

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

   11. Simplified 15.3%

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

Alternative 17: 16.6% accurate, 70.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot th}{kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (if (<= kx 6.8e-6) th (/ (* ky th) kx)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.8e-6) {
		tmp = th;
	} else {
		tmp = (ky * th) / 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 <= 6.8d-6) then
        tmp = th
    else
        tmp = (ky * th) / kx
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.8e-6) {
		tmp = th;
	} else {
		tmp = (ky * th) / kx;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 6.8e-6:
		tmp = th
	else:
		tmp = (ky * th) / kx
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 6.8e-6)
		tmp = th;
	else
		tmp = Float64(Float64(ky * th) / kx);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 6.8e-6)
		tmp = th;
	else
		tmp = (ky * th) / kx;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 6.8e-6], th, N[(N[(ky * th), $MachinePrecision] / kx), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 6.80000000000000012e-6

   1. Initial program 90.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 30.6%

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

   7. Simplified 30.6%

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

   8. Taylor expanded in th around 0

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

   10. Simplified 16.1%

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

   if 6.80000000000000012e-6 < kx

   1. Initial program 99.3%

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

   3. Taylor expanded in th around 0

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

   5. Simplified 44.5%

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

   6. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 15.2%

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

   8. Simplified 15.2%

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

   9. Taylor expanded in ky around 0

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

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

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

       2. *-lowering-*.f64 15.3%

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

   11. Simplified 15.3%

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

Alternative 18: 16.6% accurate, 70.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 3.9 \cdot 10^{-7}:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (if (<= kx 3.9e-7) th (* ky (/ th kx))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.9e-7) {
		tmp = th;
	} else {
		tmp = ky * (th / 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 <= 3.9d-7) then
        tmp = th
    else
        tmp = ky * (th / kx)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.9e-7) {
		tmp = th;
	} else {
		tmp = ky * (th / kx);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 3.9e-7:
		tmp = th
	else:
		tmp = ky * (th / kx)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 3.9e-7)
		tmp = th;
	else
		tmp = Float64(ky * Float64(th / kx));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 3.9e-7)
		tmp = th;
	else
		tmp = ky * (th / kx);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 3.9e-7], th, N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 3.90000000000000025e-7

   1. Initial program 90.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 30.6%

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

   7. Simplified 30.6%

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

   8. Taylor expanded in th around 0

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

   10. Simplified 16.1%

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

   if 3.90000000000000025e-7 < kx

   1. Initial program 99.3%

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

   3. Taylor expanded in th around 0

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

   5. Simplified 44.5%

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

   6. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 15.2%

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

   8. Simplified 15.2%

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

   9. Taylor expanded in ky around 0

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

       1. /-lowering-/.f64 15.1%

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

   11. Simplified 15.1%

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

       1. associate-*l/ N/A

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

       2. associate-/l* N/A

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

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

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

       4. /-lowering-/.f64 15.3%

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

   13. Applied egg-rr 15.3%

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

Alternative 19: 14.0% 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 92.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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 25.3%

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

7. Simplified 25.3%

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

8. Taylor expanded in th around 0

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

10. Simplified 13.6%

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

Reproduce

herbie shell --seed 2024141 
(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)))