Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.5% → 99.7%

Time: 13.0s

Alternatives: 23

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.5% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

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

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

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

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

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

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

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

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

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

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

   12. sin-lowering-sin.f64 99.6%

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

4. Applied egg-rr 99.6%

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

Alternative 2: 45.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.1:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 10^{-29}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.1)
   (/ th (/ (hypot kx (sin ky)) (sin ky)))
   (if (<= (sin ky) 1e-29) (/ (sin th) (/ (sin kx) (sin ky))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.1) {
		tmp = th / (hypot(kx, sin(ky)) / sin(ky));
	} else if (sin(ky) <= 1e-29) {
		tmp = sin(th) / (sin(kx) / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.1) {
		tmp = th / (Math.hypot(kx, Math.sin(ky)) / Math.sin(ky));
	} else if (Math.sin(ky) <= 1e-29) {
		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) <= -0.1:
		tmp = th / (math.hypot(kx, math.sin(ky)) / math.sin(ky))
	elif math.sin(ky) <= 1e-29:
		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) <= -0.1)
		tmp = Float64(th / Float64(hypot(kx, sin(ky)) / sin(ky)));
	elseif (sin(ky) <= 1e-29)
		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) <= -0.1)
		tmp = th / (hypot(kx, sin(ky)) / sin(ky));
	elseif (sin(ky) <= 1e-29)
		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], -0.1], N[(th / N[(N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-29], 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) < -0.10000000000000001

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 60.4%

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

   8. Taylor expanded in kx around 0

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

   10. Simplified 30.5%

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

   if -0.10000000000000001 < (sin.f64 ky) < 9.99999999999999943e-30

   1. Initial program 82.8%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 44.3%

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

   7. Simplified 44.3%

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

   if 9.99999999999999943e-30 < (sin.f64 ky)

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 48.4%

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

   5. Simplified 48.4%

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

Alternative 3: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

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

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

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

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

   6. sin-lowering-sin.f64 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 63.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 75000000:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(0.16666666666666666 \cdot \left(th \cdot th\right) + 1\right)}{th}}\\ \mathbf{elif}\;th \leq 4.9 \cdot 10^{+116}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 75000000.0)
   (/
    (sin ky)
    (/
     (* (hypot (sin kx) (sin ky)) (+ (* 0.16666666666666666 (* th th)) 1.0))
     th))
   (if (<= th 4.9e+116)
     (* (sin ky) (/ (sin th) (hypot kx (sin ky))))
     (/ (sin th) (/ (hypot (sin kx) ky) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 75000000.0) {
		tmp = sin(ky) / ((hypot(sin(kx), sin(ky)) * ((0.16666666666666666 * (th * th)) + 1.0)) / th);
	} else if (th <= 4.9e+116) {
		tmp = sin(ky) * (sin(th) / hypot(kx, sin(ky)));
	} else {
		tmp = sin(th) / (hypot(sin(kx), ky) / sin(ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 75000000.0) {
		tmp = Math.sin(ky) / ((Math.hypot(Math.sin(kx), Math.sin(ky)) * ((0.16666666666666666 * (th * th)) + 1.0)) / th);
	} else if (th <= 4.9e+116) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), ky) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 75000000.0:
		tmp = math.sin(ky) / ((math.hypot(math.sin(kx), math.sin(ky)) * ((0.16666666666666666 * (th * th)) + 1.0)) / th)
	elif th <= 4.9e+116:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 75000000.0)
		tmp = Float64(sin(ky) / Float64(Float64(hypot(sin(kx), sin(ky)) * Float64(Float64(0.16666666666666666 * Float64(th * th)) + 1.0)) / th));
	elseif (th <= 4.9e+116)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 75000000.0)
		tmp = sin(ky) / ((hypot(sin(kx), sin(ky)) * ((0.16666666666666666 * (th * th)) + 1.0)) / th);
	elseif (th <= 4.9e+116)
		tmp = sin(ky) * (sin(th) / hypot(kx, sin(ky)));
	else
		tmp = sin(th) / (hypot(sin(kx), ky) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 75000000.0], N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 4.9e+116], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 7.5e7

   1. Initial program 92.5%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-/r/ N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in th around 0

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

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

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

   9. Simplified 68.9%

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

   if 7.5e7 < th < 4.8999999999999998e116

   1. Initial program 90.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in kx around 0

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

   7. Simplified 61.3%

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

   if 4.8999999999999998e116 < th

   1. Initial program 82.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 59.0%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 75000000:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(0.16666666666666666 \cdot \left(th \cdot th\right) + 1\right)}{th}}\\ \mathbf{elif}\;th \leq 4.9 \cdot 10^{+116}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 11:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + th \cdot \left(th \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;th \leq 6.1 \cdot 10^{+116}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 11.0)
   (*
    (/ (sin ky) (hypot (sin ky) (sin kx)))
    (* th (+ 1.0 (* th (* th -0.16666666666666666)))))
   (if (<= th 6.1e+116)
     (* (sin ky) (/ (sin th) (hypot kx (sin ky))))
     (/ (sin th) (/ (hypot (sin kx) ky) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 11.0) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * (th * (1.0 + (th * (th * -0.16666666666666666))));
	} else if (th <= 6.1e+116) {
		tmp = sin(ky) * (sin(th) / hypot(kx, sin(ky)));
	} else {
		tmp = sin(th) / (hypot(sin(kx), ky) / sin(ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 11.0) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * (th * (1.0 + (th * (th * -0.16666666666666666))));
	} else if (th <= 6.1e+116) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), ky) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 11.0:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * (th * (1.0 + (th * (th * -0.16666666666666666))))
	elif th <= 6.1e+116:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 11.0)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * Float64(th * Float64(1.0 + Float64(th * Float64(th * -0.16666666666666666)))));
	elseif (th <= 6.1e+116)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 11.0)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * (th * (1.0 + (th * (th * -0.16666666666666666))));
	elseif (th <= 6.1e+116)
		tmp = sin(ky) * (sin(th) / hypot(kx, sin(ky)));
	else
		tmp = sin(th) / (hypot(sin(kx), ky) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 11.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(1.0 + N[(th * N[(th * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 6.1e+116], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 11

   1. Initial program 92.4%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. sin-lowering-sin.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in th around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 68.4%

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

   7. Simplified 68.4%

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

   if 11 < th < 6.10000000000000018e116

   1. Initial program 91.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in kx around 0

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

   7. Simplified 59.0%

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

   if 6.10000000000000018e116 < th

   1. Initial program 82.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 59.0%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 11:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + th \cdot \left(th \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;th \leq 6.1 \cdot 10^{+116}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 75000000:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 6.2 \cdot 10^{+116}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 75000000.0)
   (* th (/ (sin ky) (hypot (sin ky) (sin kx))))
   (if (<= th 6.2e+116)
     (* (sin ky) (/ (sin th) (hypot kx (sin ky))))
     (/ (sin th) (/ (hypot (sin kx) ky) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 75000000.0) {
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	} else if (th <= 6.2e+116) {
		tmp = sin(ky) * (sin(th) / hypot(kx, sin(ky)));
	} else {
		tmp = sin(th) / (hypot(sin(kx), ky) / sin(ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 75000000.0) {
		tmp = th * (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (th <= 6.2e+116) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), ky) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 75000000.0:
		tmp = th * (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx)))
	elif th <= 6.2e+116:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 75000000.0)
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(ky), sin(kx))));
	elseif (th <= 6.2e+116)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 75000000.0)
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	elseif (th <= 6.2e+116)
		tmp = sin(ky) * (sin(th) / hypot(kx, sin(ky)));
	else
		tmp = sin(th) / (hypot(sin(kx), ky) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 75000000.0], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 6.2e+116], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 7.5e7

   1. Initial program 92.5%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 68.5%

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

   if 7.5e7 < th < 6.19999999999999992e116

   1. Initial program 90.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in kx around 0

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

   7. Simplified 61.3%

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

   if 6.19999999999999992e116 < th

   1. Initial program 82.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 59.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 75000000:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 6.2 \cdot 10^{+116}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 75000000:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 5.5 \cdot 10^{+116}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 75000000.0)
   (* th (/ (sin ky) (hypot (sin ky) (sin kx))))
   (if (<= th 5.5e+116)
     (* (sin ky) (/ (sin th) (hypot kx (sin ky))))
     (/ (sin ky) (/ (hypot (sin kx) ky) (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 75000000.0) {
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	} else if (th <= 5.5e+116) {
		tmp = sin(ky) * (sin(th) / hypot(kx, sin(ky)));
	} else {
		tmp = sin(ky) / (hypot(sin(kx), ky) / sin(th));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 75000000.0) {
		tmp = th * (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (th <= 5.5e+116) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), ky) / Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 75000000.0:
		tmp = th * (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx)))
	elif th <= 5.5e+116:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), ky) / math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 75000000.0)
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(ky), sin(kx))));
	elseif (th <= 5.5e+116)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), ky) / sin(th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 75000000.0)
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	elseif (th <= 5.5e+116)
		tmp = sin(ky) * (sin(th) / hypot(kx, sin(ky)));
	else
		tmp = sin(ky) / (hypot(sin(kx), ky) / sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 75000000.0], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 5.5e+116], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 7.5e7

   1. Initial program 92.5%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 68.5%

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

   if 7.5e7 < th < 5.50000000000000035e116

   1. Initial program 90.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in kx around 0

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

   7. Simplified 61.3%

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

   if 5.50000000000000035e116 < th

   1. Initial program 82.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. associate-/r/ N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in ky around 0

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

   9. Simplified 59.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 75000000:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 5.5 \cdot 10^{+116}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 75000000:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 3.4 \cdot 10^{+116}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 75000000.0)
   (* th (/ (sin ky) (hypot (sin ky) (sin kx))))
   (if (<= th 3.4e+116)
     (* (sin ky) (/ (sin th) (hypot kx (sin ky))))
     (* (sin th) (/ (sin ky) (hypot ky (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 75000000.0) {
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	} else if (th <= 3.4e+116) {
		tmp = sin(ky) * (sin(th) / hypot(kx, sin(ky)));
	} else {
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 75000000.0) {
		tmp = th * (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (th <= 3.4e+116) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 75000000.0:
		tmp = th * (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx)))
	elif th <= 3.4e+116:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 75000000.0)
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(ky), sin(kx))));
	elseif (th <= 3.4e+116)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 75000000.0)
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	elseif (th <= 3.4e+116)
		tmp = sin(ky) * (sin(th) / hypot(kx, sin(ky)));
	else
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 75000000.0], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 3.4e+116], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 7.5e7

   1. Initial program 92.5%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 68.5%

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

   if 7.5e7 < th < 3.40000000000000023e116

   1. Initial program 90.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in kx around 0

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

   7. Simplified 61.3%

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

   if 3.40000000000000023e116 < th

   1. Initial program 82.6%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. sin-lowering-sin.f64 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 59.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 75000000:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 3.4 \cdot 10^{+116}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.0025:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.0025)
   (* th (/ (sin ky) (hypot (sin ky) (sin kx))))
   (* (sin th) (/ (sin ky) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0025) {
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0025) {
		tmp = th * (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.0025:
		tmp = th * (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.0025)
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.0025)
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.0025], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.00250000000000000005

   1. Initial program 92.4%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. sin-lowering-sin.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 68.8%

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

   if 0.00250000000000000005 < th

   1. Initial program 85.7%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 53.9%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.0025:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.00031:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.00031)
   (* (sin ky) (/ th (hypot (sin kx) (sin ky))))
   (* (sin th) (/ (sin ky) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.00031) {
		tmp = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	} else {
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.00031) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(kx), Math.sin(ky)));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.00031:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(kx), math.sin(ky)))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.00031)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(kx), sin(ky))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.00031)
		tmp = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	else
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.00031], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 3.1e-4

   1. Initial program 92.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 68.7%

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

   if 3.1e-4 < th

   1. Initial program 85.7%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 53.9%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.00031:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 75000000:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{elif}\;th \leq 1.16 \cdot 10^{+73}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 75000000.0)
   (* (sin ky) (/ th (hypot (sin kx) (sin ky))))
   (if (<= th 1.16e+73) (sin th) (/ (sin th) (/ (sin kx) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 75000000.0) {
		tmp = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	} else if (th <= 1.16e+73) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(kx) / sin(ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 75000000.0) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(kx), Math.sin(ky)));
	} else if (th <= 1.16e+73) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 75000000.0:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(kx), math.sin(ky)))
	elif th <= 1.16e+73:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 75000000.0)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(kx), sin(ky))));
	elseif (th <= 1.16e+73)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 75000000.0)
		tmp = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	elseif (th <= 1.16e+73)
		tmp = sin(th);
	else
		tmp = sin(th) / (sin(kx) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 75000000.0], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 1.16e+73], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 7.5e7

   1. Initial program 92.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 68.4%

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

   if 7.5e7 < th < 1.16000000000000007e73

   1. Initial program 89.9%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 33.1%

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

   5. Simplified 33.1%

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

   if 1.16000000000000007e73 < th

   1. Initial program 84.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 18.0%

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

   7. Simplified 18.0%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 75000000:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{elif}\;th \leq 1.16 \cdot 10^{+73}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-29}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-29) (/ (sin th) (/ (sin kx) (sin ky))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-29) {
		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) <= 1d-29) 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) <= 1e-29) {
		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) <= 1e-29:
		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) <= 1e-29)
		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) <= 1e-29)
		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], 1e-29], 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 2 regimes

2. if (sin.f64 ky) < 9.99999999999999943e-30

   1. Initial program 87.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 33.8%

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

   7. Simplified 33.8%

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

   if 9.99999999999999943e-30 < (sin.f64 ky)

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 48.4%

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

   5. Simplified 48.4%

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

Alternative 13: 41.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-29}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-29) (* (sin th) (/ (sin ky) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-29) {
		tmp = sin(th) * (sin(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) <= 1d-29) then
        tmp = sin(th) * (sin(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) <= 1e-29) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-29:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-29)
		tmp = Float64(sin(th) * Float64(sin(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) <= 1e-29)
		tmp = sin(th) * (sin(ky) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-29], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.99999999999999943e-30

   1. Initial program 87.7%

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

   3. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 33.8%

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

   5. Simplified 33.8%

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

   if 9.99999999999999943e-30 < (sin.f64 ky)

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 48.4%

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

   5. Simplified 48.4%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-29}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 40.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-29}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx \cdot \left(1 + 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-29)
   (/ (sin th) (/ (* (sin kx) (+ 1.0 (* 0.16666666666666666 (* ky ky)))) ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-29) {
		tmp = sin(th) / ((sin(kx) * (1.0 + (0.16666666666666666 * (ky * ky)))) / 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) <= 1d-29) then
        tmp = sin(th) / ((sin(kx) * (1.0d0 + (0.16666666666666666d0 * (ky * ky)))) / 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) <= 1e-29) {
		tmp = Math.sin(th) / ((Math.sin(kx) * (1.0 + (0.16666666666666666 * (ky * ky)))) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-29:
		tmp = math.sin(th) / ((math.sin(kx) * (1.0 + (0.16666666666666666 * (ky * ky)))) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-29)
		tmp = Float64(sin(th) / Float64(Float64(sin(kx) * Float64(1.0 + Float64(0.16666666666666666 * Float64(ky * ky)))) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-29)
		tmp = sin(th) / ((sin(kx) * (1.0 + (0.16666666666666666 * (ky * ky)))) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-29], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[Sin[kx], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.99999999999999943e-30

   1. Initial program 87.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 33.8%

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

   7. Simplified 33.8%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt1-in N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

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

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

      15. sin-lowering-sin.f64 33.1%

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

   10. Simplified 33.1%

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

   if 9.99999999999999943e-30 < (sin.f64 ky)

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 48.4%

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

   5. Simplified 48.4%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-29}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx \cdot \left(1 + 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 33.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-88}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 4e-88) (* (sin th) (/ (sin ky) kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 4e-88) {
		tmp = sin(th) * (sin(ky) / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 4d-88) then
        tmp = sin(th) * (sin(ky) / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 4e-88) {
		tmp = Math.sin(th) * (Math.sin(ky) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 4e-88:
		tmp = math.sin(th) * (math.sin(ky) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 4e-88)
		tmp = Float64(sin(th) * Float64(sin(ky) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 4e-88)
		tmp = sin(th) * (sin(ky) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-88], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 3.99999999999999974e-88

   1. Initial program 86.9%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 34.0%

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

   7. Simplified 34.0%

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

   8. Taylor expanded in kx around 0

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

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

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 16.9%

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

   10. Simplified 16.9%

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

   11. Taylor expanded in kx around 0

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

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

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

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

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

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

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

       6. sin-lowering-sin.f64 24.0%

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

   13. Simplified 24.0%

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

   if 3.99999999999999974e-88 < (sin.f64 ky)

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 45.3%

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

   5. Simplified 45.3%

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

Alternative 16: 33.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-88}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 4e-88) (* (sin ky) (/ (sin th) kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 4e-88) {
		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) <= 4d-88) 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) <= 4e-88) {
		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) <= 4e-88:
		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) <= 4e-88)
		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) <= 4e-88)
		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], 4e-88], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 3.99999999999999974e-88

   1. Initial program 86.9%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 34.0%

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

   7. Simplified 34.0%

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

   8. Taylor expanded in kx around 0

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

      1. associate-/l* N/A

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

      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 24.0%

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

   10. Simplified 24.0%

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

   if 3.99999999999999974e-88 < (sin.f64 ky)

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 45.3%

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

   5. Simplified 45.3%

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

Alternative 17: 33.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-88}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx \cdot \left(1 + -0.16666666666666666 \cdot \left(kx \cdot kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 4e-88)
   (* (sin th) (/ ky (* kx (+ 1.0 (* -0.16666666666666666 (* kx kx))))))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 4e-88) {
		tmp = sin(th) * (ky / (kx * (1.0 + (-0.16666666666666666 * (kx * kx)))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 4d-88) then
        tmp = sin(th) * (ky / (kx * (1.0d0 + ((-0.16666666666666666d0) * (kx * kx)))))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 4e-88) {
		tmp = Math.sin(th) * (ky / (kx * (1.0 + (-0.16666666666666666 * (kx * kx)))));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 4e-88:
		tmp = math.sin(th) * (ky / (kx * (1.0 + (-0.16666666666666666 * (kx * kx)))))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 4e-88)
		tmp = Float64(sin(th) * Float64(ky / Float64(kx * Float64(1.0 + Float64(-0.16666666666666666 * Float64(kx * kx))))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 4e-88)
		tmp = sin(th) * (ky / (kx * (1.0 + (-0.16666666666666666 * (kx * kx)))));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-88], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[(kx * N[(1.0 + N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 3.99999999999999974e-88

   1. Initial program 86.9%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{ky}{\sin kx}\right)}, \mathsf{sin.f64}\left(th\right)\right) \]
   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 32.7%

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

   5. Simplified 32.7%

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

   6. Taylor expanded in kx around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 23.9%

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

   8. Simplified 23.9%

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

   if 3.99999999999999974e-88 < (sin.f64 ky)

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 45.3%

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

   5. Simplified 45.3%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-88}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx \cdot \left(1 + -0.16666666666666666 \cdot \left(kx \cdot kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 32.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.4e-29) (/ (sin th) (/ (sin kx) ky)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.4e-29) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 1.4d-29) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.4e-29) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.4e-29:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.4e-29)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.4e-29)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.4e-29], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.4000000000000001e-29

   1. Initial program 88.0%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in ky around 0

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

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

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

      2. sin-lowering-sin.f64 31.7%

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

   7. Simplified 31.7%

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

   if 1.4000000000000001e-29 < ky

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 35.6%

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

   5. Simplified 35.6%

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

Alternative 19: 32.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 10^{-29}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1e-29) (* (sin th) (/ ky (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1e-29) {
		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 (ky <= 1d-29) 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 (ky <= 1e-29) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1e-29:
		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 (ky <= 1e-29)
		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 (ky <= 1e-29)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1e-29], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 9.99999999999999943e-30

   1. Initial program 88.0%

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

   3. Taylor expanded in ky around 0

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

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

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

      2. sin-lowering-sin.f64 31.7%

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

   5. Simplified 31.7%

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

   if 9.99999999999999943e-30 < ky

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 35.6%

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

   5. Simplified 35.6%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 10^{-29}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2.3 \cdot 10^{-88}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 2.3e-88) (* (sin th) (/ ky kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.3e-88) {
		tmp = sin(th) * (ky / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 2.3d-88) then
        tmp = sin(th) * (ky / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.3e-88) {
		tmp = Math.sin(th) * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 2.3e-88:
		tmp = math.sin(th) * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 2.3e-88)
		tmp = Float64(sin(th) * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 2.3e-88)
		tmp = sin(th) * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 2.3e-88], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.29999999999999986e-88

   1. Initial program 87.3%

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

   3. Taylor expanded in ky around 0

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

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

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

      2. sin-lowering-sin.f64 31.8%

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

   5. Simplified 31.8%

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

   6. Taylor expanded in kx around 0

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

      1. /-lowering-/.f64 23.2%

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

   8. Simplified 23.2%

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

   if 2.29999999999999986e-88 < ky

   1. Initial program 99.7%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 34.2%

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

   5. Simplified 34.2%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 2.3 \cdot 10^{-88}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 24.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.45 \cdot 10^{+38}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.45e+38) (sin th) (* -0.16666666666666666 (* th (* th th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.45e+38) {
		tmp = sin(th);
	} else {
		tmp = -0.16666666666666666 * (th * (th * th));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 1.45d+38) then
        tmp = sin(th)
    else
        tmp = (-0.16666666666666666d0) * (th * (th * th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.45e+38) {
		tmp = Math.sin(th);
	} else {
		tmp = -0.16666666666666666 * (th * (th * th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.45e+38:
		tmp = math.sin(th)
	else:
		tmp = -0.16666666666666666 * (th * (th * th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.45e+38)
		tmp = sin(th);
	else
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.45e+38)
		tmp = sin(th);
	else
		tmp = -0.16666666666666666 * (th * (th * th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.45e+38], N[Sin[th], $MachinePrecision], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.45000000000000003e38

   1. Initial program 88.4%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 20.9%

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

   5. Simplified 20.9%

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

   if 1.45000000000000003e38 < kx

   1. Initial program 99.2%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 9.2%

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

   5. Simplified 9.2%

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

   6. Taylor expanded in th around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \color{blue}{\left(th \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 6.0%

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 6.0%

      \[\leadsto \color{blue}{th \cdot \left(1 + th \cdot \left(th \cdot -0.16666666666666666\right)\right)} \]

   9. Taylor expanded in th around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({th}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \left(th \cdot {th}^{\color{blue}{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, \color{blue}{\left({th}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, \left(th \cdot \color{blue}{th}\right)\right)\right) \]

       6. *-lowering-*.f64 9.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \color{blue}{th}\right)\right)\right) \]

   11. Simplified 9.3%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 15.9% accurate, 59.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 12000:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 12000.0) th (* -0.16666666666666666 (* th (* th th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 12000.0) {
		tmp = th;
	} else {
		tmp = -0.16666666666666666 * (th * (th * th));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 12000.0d0) then
        tmp = th
    else
        tmp = (-0.16666666666666666d0) * (th * (th * th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 12000.0) {
		tmp = th;
	} else {
		tmp = -0.16666666666666666 * (th * (th * th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 12000.0:
		tmp = th
	else:
		tmp = -0.16666666666666666 * (th * (th * th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 12000.0)
		tmp = th;
	else
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 12000.0)
		tmp = th;
	else
		tmp = -0.16666666666666666 * (th * (th * th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 12000.0], th, N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 12000

   1. Initial program 88.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 21.4%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   5. Simplified 21.4%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th} \]
   7. Step-by-step derivation

   8. Simplified 12.1%

      \[\leadsto \color{blue}{th} \]

   if 12000 < 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 kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 8.9%

         \[\leadsto \mathsf{sin.f64}\left(th\right) \]

   5. Simplified 8.9%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(th, \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left({th}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(\left(th \cdot th\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \left(th \cdot \color{blue}{\left(th \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \color{blue}{\left(th \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 5.8%

         \[\leadsto \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 5.8%

      \[\leadsto \color{blue}{th \cdot \left(1 + th \cdot \left(th \cdot -0.16666666666666666\right)\right)} \]

   9. Taylor expanded in th around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({th}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \left(th \cdot {th}^{\color{blue}{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, \color{blue}{\left({th}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, \left(th \cdot \color{blue}{th}\right)\right)\right) \]

       6. *-lowering-*.f64 9.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \color{blue}{th}\right)\right)\right) \]

   11. Simplified 9.3%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 13.7% 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 90.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 18.4%

      \[\leadsto \mathsf{sin.f64}\left(th\right) \]

5. Simplified 18.4%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto \color{blue}{th} \]
7. Step-by-step derivation

8. Simplified 10.6%

   \[\leadsto \color{blue}{th} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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)))