Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 12.4s

Alternatives: 17

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

3. Taylor expanded in kx around inf

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. hypot-define N/A

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

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

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

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

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

   6. sin-lowering-sin.f64 99.6%

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

5. Simplified 99.6%

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

Alternative 2: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. hypot-define N/A

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

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

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

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

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

   13. sin-lowering-sin.f64 99.6%

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

3. Simplified 99.6%

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

Alternative 3: 67.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.0069:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \left(th \cdot \left(1 + th \cdot \left(th \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.0069)
   (*
    (/ (sin ky) (hypot (sin kx) (sin ky)))
    (* th (+ 1.0 (* th (* th -0.16666666666666666)))))
   (* ky (/ (sin th) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0069) {
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * (th * (1.0 + (th * (th * -0.16666666666666666))));
	} else {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0069) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky))) * (th * (1.0 + (th * (th * -0.16666666666666666))));
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.0069:
		tmp = (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky))) * (th * (1.0 + (th * (th * -0.16666666666666666))))
	else:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.0069)
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * Float64(th * Float64(1.0 + Float64(th * Float64(th * -0.16666666666666666)))));
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.0069)
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * (th * (1.0 + (th * (th * -0.16666666666666666))));
	else
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.0069], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(1.0 + N[(th * N[(th * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.0068999999999999999

   1. Initial program 93.4%

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

   3. Taylor expanded in kx around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. hypot-define N/A

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

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

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

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

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

      6. sin-lowering-sin.f64 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in th around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 61.8%

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

   8. Simplified 61.8%

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

   if 0.0068999999999999999 < th

   1. Initial program 91.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 67.5%

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

   8. Taylor expanded in ky around 0

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

   10. Simplified 75.4%

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

Alternative 4: 67.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.0057:\\ \;\;\;\;\sin ky \cdot \frac{th \cdot \left(1 + th \cdot \left(th \cdot -0.16666666666666666\right)\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.0057)
   (*
    (sin ky)
    (/
     (* th (+ 1.0 (* th (* th -0.16666666666666666))))
     (hypot (sin ky) (sin kx))))
   (* ky (/ (sin th) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0057) {
		tmp = sin(ky) * ((th * (1.0 + (th * (th * -0.16666666666666666)))) / hypot(sin(ky), sin(kx)));
	} else {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0057) {
		tmp = Math.sin(ky) * ((th * (1.0 + (th * (th * -0.16666666666666666)))) / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.0057:
		tmp = math.sin(ky) * ((th * (1.0 + (th * (th * -0.16666666666666666)))) / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.0057)
		tmp = Float64(sin(ky) * Float64(Float64(th * Float64(1.0 + Float64(th * Float64(th * -0.16666666666666666)))) / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.0057)
		tmp = sin(ky) * ((th * (1.0 + (th * (th * -0.16666666666666666)))) / hypot(sin(ky), sin(kx)));
	else
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.0057], N[(N[Sin[ky], $MachinePrecision] * N[(N[(th * N[(1.0 + N[(th * N[(th * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.0057000000000000002

   1. Initial program 93.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

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

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

      6. *-lowering-*.f64 61.7%

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

   7. Simplified 61.7%

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

   if 0.0057000000000000002 < th

   1. Initial program 91.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 67.5%

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

   8. Taylor expanded in ky around 0

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

   10. Simplified 75.4%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.0057:\\ \;\;\;\;\sin ky \cdot \frac{th \cdot \left(1 + th \cdot \left(th \cdot -0.16666666666666666\right)\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.05:\\ \;\;\;\;th \cdot \frac{\frac{1}{\frac{1}{\sin ky}}}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.05)
   (* th (/ (/ 1.0 (/ 1.0 (sin ky))) (hypot kx (sin ky))))
   (* ky (/ (sin th) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.05) {
		tmp = th * ((1.0 / (1.0 / sin(ky))) / hypot(kx, sin(ky)));
	} else {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.05) {
		tmp = th * ((1.0 / (1.0 / Math.sin(ky))) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.05:
		tmp = th * ((1.0 / (1.0 / math.sin(ky))) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.05)
		tmp = Float64(th * Float64(Float64(1.0 / Float64(1.0 / sin(ky))) / hypot(kx, sin(ky))));
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.05)
		tmp = th * ((1.0 / (1.0 / sin(ky))) / hypot(kx, sin(ky)));
	else
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.05], N[(th * N[(N[(1.0 / N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < -0.050000000000000003

   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 inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. hypot-define N/A

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

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

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

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

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

      6. sin-lowering-sin.f64 99.6%

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

   5. Simplified 99.6%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

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

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

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

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

      5. sin-lowering-sin.f64 99.3%

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in th around 0

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

   10. Simplified 36.7%

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

   11. Taylor expanded in kx around 0

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

   13. Simplified 15.0%

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

   if -0.050000000000000003 < (sin.f64 ky)

   1. Initial program 91.0%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 78.2%

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

   8. Taylor expanded in ky around 0

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

   10. Simplified 83.5%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.05:\\ \;\;\;\;th \cdot \frac{\frac{1}{\frac{1}{\sin ky}}}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.00235:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.00235)
   (* (/ (sin ky) (hypot (sin kx) (sin ky))) th)
   (* ky (/ (sin th) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.00235) {
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	} else {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.00235) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky))) * th;
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.00235:
		tmp = (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky))) * th
	else:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.00235)
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * th);
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.00235)
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	else
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.00235], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.00235000000000000009

   1. Initial program 93.4%

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

   3. Taylor expanded in kx around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. hypot-define N/A

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

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

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

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

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

      6. sin-lowering-sin.f64 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in th around 0

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

   8. Simplified 62.2%

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

   if 0.00235000000000000009 < th

   1. Initial program 91.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 67.5%

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

   8. Taylor expanded in ky around 0

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

   10. Simplified 75.4%

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

Alternative 7: 67.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.000122:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.000122)
   (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
   (* ky (/ (sin th) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.000122) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.000122) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.000122:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.000122)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.000122)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	else
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.000122], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1.21999999999999997e-4

   1. Initial program 93.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 62.1%

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

   if 1.21999999999999997e-4 < th

   1. Initial program 91.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 67.5%

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

   8. Taylor expanded in ky around 0

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

   10. Simplified 75.4%

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

Alternative 8: 40.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq 10^{-104}:\\ \;\;\;\;0.5 \cdot \frac{ky \cdot \left(\sin th + \sin th\right)}{ky}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) 1e-104)
   (* 0.5 (/ (* ky (+ (sin th) (sin th))) ky))
   (* (sin th) (/ (sin ky) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= 1e-104) {
		tmp = 0.5 * ((ky * (sin(th) + sin(th))) / ky);
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= 1d-104) then
        tmp = 0.5d0 * ((ky * (sin(th) + sin(th))) / ky)
    else
        tmp = sin(th) * (sin(ky) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= 1e-104) {
		tmp = 0.5 * ((ky * (Math.sin(th) + Math.sin(th))) / ky);
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= 1e-104:
		tmp = 0.5 * ((ky * (math.sin(th) + math.sin(th))) / ky)
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= 1e-104)
		tmp = Float64(0.5 * Float64(Float64(ky * Float64(sin(th) + sin(th))) / ky));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= 1e-104)
		tmp = 0.5 * ((ky * (sin(th) + sin(th))) / ky);
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-104], N[(0.5 * N[(N[(ky * N[(N[Sin[th], $MachinePrecision] + N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 kx) < 9.99999999999999927e-105

   1. Initial program 89.9%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 26.8%

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

   7. Simplified 26.8%

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

      1. *-rgt-identity N/A

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

      2. lft-mult-inverse N/A

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

      3. associate-*l* N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. sin-mult N/A

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

      9. frac-times N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(ky, th\right)\right)\right), 1\right), \mathsf{*.f64}\left(2, \color{blue}{\sin ky}\right)\right) \]

      18. sin-lowering-sin.f64 10.8%

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

   9. Applied egg-rr 10.8%

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

   10. Taylor expanded in ky around 0

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

       1. distribute-lft-out N/A

          \[\leadsto \frac{\frac{1}{2} \cdot \left(ky \cdot \left(\sin th - \sin \left(\mathsf{neg}\left(th\right)\right)\right) + \left(\cos \left(\mathsf{neg}\left(th\right)\right) - \cos th\right)\right)}{ky} \]

       2. associate-/l* N/A

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

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

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

       4. cos-neg N/A

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

       5. +-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{ky \cdot \left(\sin th - \sin \left(\mathsf{neg}\left(th\right)\right)\right) + 0}{ky}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(ky \cdot \left(\sin th - \sin \left(\mathsf{neg}\left(th\right)\right)\right) + 0\right), \color{blue}{ky}\right)\right) \]

   12. Simplified 31.6%

       \[\leadsto \color{blue}{0.5 \cdot \frac{ky \cdot \left(\sin th + \sin th\right) + 0}{ky}} \]

   if 9.99999999999999927e-105 < (sin.f64 kx)

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 62.2%

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

   5. Simplified 62.2%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq 10^{-104}:\\ \;\;\;\;0.5 \cdot \frac{ky \cdot \left(\sin th + \sin th\right)}{ky}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq 10^{-104}:\\ \;\;\;\;0.5 \cdot \frac{ky \cdot \left(\sin th + \sin th\right)}{ky}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) 1e-104)
   (* 0.5 (/ (* ky (+ (sin th) (sin th))) ky))
   (* (sin ky) (/ (sin th) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= 1e-104) {
		tmp = 0.5 * ((ky * (sin(th) + sin(th))) / ky);
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= 1d-104) then
        tmp = 0.5d0 * ((ky * (sin(th) + sin(th))) / ky)
    else
        tmp = sin(ky) * (sin(th) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= 1e-104) {
		tmp = 0.5 * ((ky * (Math.sin(th) + Math.sin(th))) / ky);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= 1e-104:
		tmp = 0.5 * ((ky * (math.sin(th) + math.sin(th))) / ky)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= 1e-104)
		tmp = Float64(0.5 * Float64(Float64(ky * Float64(sin(th) + sin(th))) / ky));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= 1e-104)
		tmp = 0.5 * ((ky * (sin(th) + sin(th))) / ky);
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-104], N[(0.5 * N[(N[(ky * N[(N[Sin[th], $MachinePrecision] + N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 kx) < 9.99999999999999927e-105

   1. Initial program 89.9%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 26.8%

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

   7. Simplified 26.8%

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

      1. *-rgt-identity N/A

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

      2. lft-mult-inverse N/A

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

      3. associate-*l* N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. sin-mult N/A

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

      9. frac-times N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(ky, th\right)\right)\right), 1\right), \mathsf{*.f64}\left(2, \color{blue}{\sin ky}\right)\right) \]

      18. sin-lowering-sin.f64 10.8%

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

   9. Applied egg-rr 10.8%

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

   10. Taylor expanded in ky around 0

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

       1. distribute-lft-out N/A

          \[\leadsto \frac{\frac{1}{2} \cdot \left(ky \cdot \left(\sin th - \sin \left(\mathsf{neg}\left(th\right)\right)\right) + \left(\cos \left(\mathsf{neg}\left(th\right)\right) - \cos th\right)\right)}{ky} \]

       2. associate-/l* N/A

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

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

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

       4. cos-neg N/A

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

       5. +-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{ky \cdot \left(\sin th - \sin \left(\mathsf{neg}\left(th\right)\right)\right) + 0}{ky}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(ky \cdot \left(\sin th - \sin \left(\mathsf{neg}\left(th\right)\right)\right) + 0\right), \color{blue}{ky}\right)\right) \]

   12. Simplified 31.6%

       \[\leadsto \color{blue}{0.5 \cdot \frac{ky \cdot \left(\sin th + \sin th\right) + 0}{ky}} \]

   if 9.99999999999999927e-105 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 62.2%

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

   7. Simplified 62.2%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq 10^{-104}:\\ \;\;\;\;0.5 \cdot \frac{ky \cdot \left(\sin th + \sin th\right)}{ky}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. hypot-define N/A

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

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

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

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

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

   13. sin-lowering-sin.f64 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in ky around 0

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

7. Simplified 62.1%

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

8. Taylor expanded in ky around 0

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

10. Simplified 70.3%

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

Alternative 11: 33.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 4.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{ky \cdot \left(\sin th + \sin th\right)}{ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 4.2e-97)
   (/ (sin th) (/ (sin kx) ky))
   (* 0.5 (/ (* ky (+ (sin th) (sin th))) ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.2e-97) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = 0.5 * ((ky * (sin(th) + sin(th))) / ky);
	}
	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 <= 4.2d-97) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = 0.5d0 * ((ky * (sin(th) + sin(th))) / ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.2e-97) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = 0.5 * ((ky * (Math.sin(th) + Math.sin(th))) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 4.2e-97:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = 0.5 * ((ky * (math.sin(th) + math.sin(th))) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 4.2e-97)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = Float64(0.5 * Float64(Float64(ky * Float64(sin(th) + sin(th))) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 4.2e-97)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = 0.5 * ((ky * (sin(th) + sin(th))) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 4.2e-97], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(ky * N[(N[Sin[th], $MachinePrecision] + N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 4.2000000000000002e-97

   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 inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. hypot-define N/A

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

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

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

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

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

      6. sin-lowering-sin.f64 99.6%

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

   5. Simplified 99.6%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

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

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

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

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

      5. sin-lowering-sin.f64 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. hypot-define N/A

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

      7. remove-double-div N/A

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

      8. hypot-define N/A

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

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

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in ky around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \color{blue}{\left(\frac{\sin kx}{ky}\right)}\right) \]
   11. 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 33.4%

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

   12. Simplified 33.4%

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

   if 4.2000000000000002e-97 < ky

   1. Initial program 99.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 36.0%

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

   7. Simplified 36.0%

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

      1. *-rgt-identity N/A

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

      2. lft-mult-inverse N/A

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

      3. associate-*l* N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. sin-mult N/A

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

      9. frac-times N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(ky, th\right)\right)\right), 1\right), \mathsf{*.f64}\left(2, \color{blue}{\sin ky}\right)\right) \]

      18. sin-lowering-sin.f64 10.1%

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

   9. Applied egg-rr 10.1%

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

   10. Taylor expanded in ky around 0

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

       1. distribute-lft-out N/A

          \[\leadsto \frac{\frac{1}{2} \cdot \left(ky \cdot \left(\sin th - \sin \left(\mathsf{neg}\left(th\right)\right)\right) + \left(\cos \left(\mathsf{neg}\left(th\right)\right) - \cos th\right)\right)}{ky} \]

       2. associate-/l* N/A

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

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

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

       4. cos-neg N/A

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

       5. +-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{ky \cdot \left(\sin th - \sin \left(\mathsf{neg}\left(th\right)\right)\right) + 0}{ky}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(ky \cdot \left(\sin th - \sin \left(\mathsf{neg}\left(th\right)\right)\right) + 0\right), \color{blue}{ky}\right)\right) \]

   12. Simplified 40.5%

       \[\leadsto \color{blue}{0.5 \cdot \frac{ky \cdot \left(\sin th + \sin th\right) + 0}{ky}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 4.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{ky \cdot \left(\sin th + \sin th\right)}{ky}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.65 \cdot 10^{-104}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.65e-104) (sin th) (* (sin th) (/ ky (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.65e-104) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (ky / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 1.65d-104) then
        tmp = sin(th)
    else
        tmp = sin(th) * (ky / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.65e-104) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.65e-104:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.65e-104)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.65e-104)
		tmp = sin(th);
	else
		tmp = sin(th) * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.65e-104], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.65000000000000001e-104

   1. Initial program 89.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 27.8%

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

   7. Simplified 27.8%

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

   if 1.65000000000000001e-104 < kx

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

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

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

      2. sin-lowering-sin.f64 42.4%

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

   5. Simplified 42.4%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.65 \cdot 10^{-104}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 380000:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 380000.0) (sin th) (* th (/ ky (sin kx)))))

C

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

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 380000.0d0) then
        tmp = sin(th)
    else
        tmp = th * (ky / sin(kx))
    end if
    code = tmp
end function

Java

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

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 380000.0:
		tmp = math.sin(th)
	else:
		tmp = th * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 380000.0)
		tmp = sin(th);
	else
		tmp = Float64(th * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 380000.0)
		tmp = sin(th);
	else
		tmp = th * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 380000.0], N[Sin[th], $MachinePrecision], N[(th * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 3.8e5

   1. Initial program 90.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 27.6%

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

   7. Simplified 27.6%

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

   if 3.8e5 < kx

   1. Initial program 99.4%

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

   3. Taylor expanded in th around 0

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

   5. Simplified 47.8%

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

   6. Taylor expanded in ky around 0

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

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

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

      2. sin-lowering-sin.f64 22.4%

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

   8. Simplified 22.4%

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

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 380000:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 180000:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 180000.0) (sin th) (* ky (/ th (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 180000.0) {
		tmp = sin(th);
	} else {
		tmp = ky * (th / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 180000.0d0) then
        tmp = sin(th)
    else
        tmp = ky * (th / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 180000.0) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (th / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 180000.0:
		tmp = math.sin(th)
	else:
		tmp = ky * (th / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 180000.0)
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(th / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 180000.0)
		tmp = sin(th);
	else
		tmp = ky * (th / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 180000.0], N[Sin[th], $MachinePrecision], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.8e5

   1. Initial program 90.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 27.6%

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

   7. Simplified 27.6%

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

   if 1.8e5 < kx

   1. Initial program 99.4%

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

   3. Taylor expanded in th around 0

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

   5. Simplified 47.8%

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

   6. Taylor expanded in ky around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 22.4%

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

   8. Simplified 22.4%

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

Alternative 15: 25.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 3.15 \cdot 10^{+25}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (if (<= kx 3.15e+25) (sin th) 0.0))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.15e+25) {
		tmp = sin(th);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 3.15d+25) then
        tmp = sin(th)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.15e+25) {
		tmp = Math.sin(th);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 3.15e+25:
		tmp = math.sin(th)
	else:
		tmp = 0.0
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 3.15e+25)
		tmp = sin(th);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 3.15e+25)
		tmp = sin(th);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 3.15e+25], N[Sin[th], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if kx < 3.14999999999999987e25

   1. Initial program 90.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 27.0%

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

   7. Simplified 27.0%

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

   if 3.14999999999999987e25 < kx

   1. Initial program 99.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 6.8%

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

   7. Simplified 6.8%

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

      1. *-rgt-identity N/A

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

      2. lft-mult-inverse N/A

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

      3. associate-*l* N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. sin-mult N/A

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

      9. frac-times N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(ky, th\right)\right)\right), 1\right), \mathsf{*.f64}\left(2, \color{blue}{\sin ky}\right)\right) \]

      18. sin-lowering-sin.f64 21.7%

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

   9. Applied egg-rr 21.7%

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

   10. Taylor expanded in ky around 0

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

       1. cos-neg N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\cos \left(\mathsf{neg}\left(th\right)\right) - \cos \left(\mathsf{neg}\left(th\right)\right)}{ky} \]

       2. div-sub N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{\cos \left(\mathsf{neg}\left(th\right)\right)}{ky} - \color{blue}{\frac{\cos \left(\mathsf{neg}\left(th\right)\right)}{ky}}\right) \]

       3. +-inverses N/A

          \[\leadsto \frac{1}{2} \cdot 0 \]

       4. metadata-eval 21.5%

          \[\leadsto 0 \]

   12. Simplified 21.5%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 16.2% accurate, 117.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 7 \cdot 10^{-38}:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (if (<= kx 7e-38) th 0.0))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 7e-38) {
		tmp = th;
	} else {
		tmp = 0.0;
	}
	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 <= 7d-38) then
        tmp = th
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 7e-38) {
		tmp = th;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 7e-38:
		tmp = th
	else:
		tmp = 0.0
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 7e-38)
		tmp = th;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 7e-38)
		tmp = th;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 7e-38], th, 0.0]

Derivation

1. Split input into 2 regimes

2. if kx < 7.0000000000000003e-38

   1. Initial program 89.9%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 28.3%

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

   7. Simplified 28.3%

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

   8. Taylor expanded in th around 0

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

   10. Simplified 19.8%

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

   if 7.0000000000000003e-38 < kx

   1. Initial program 99.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 7.6%

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

   7. Simplified 7.6%

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

      1. *-rgt-identity N/A

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

      2. lft-mult-inverse N/A

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

      3. associate-*l* N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \color{blue}{\frac{1}{\sin ky}} \]

      8. sin-mult N/A

         \[\leadsto \frac{\cos \left(ky - th\right) - \cos \left(ky + th\right)}{2} \cdot \frac{\color{blue}{1}}{\sin ky} \]

      9. frac-times N/A

         \[\leadsto \frac{\left(\cos \left(ky - th\right) - \cos \left(ky + th\right)\right) \cdot 1}{\color{blue}{2 \cdot \sin ky}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\cos \left(ky - th\right) - \cos \left(ky + th\right)\right) \cdot 1\right), \color{blue}{\left(2 \cdot \sin ky\right)}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\cos \left(ky - th\right) - \cos \left(ky + th\right)\right), 1\right), \left(\color{blue}{2} \cdot \sin ky\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\cos \left(ky - th\right), \cos \left(ky + th\right)\right), 1\right), \left(2 \cdot \sin ky\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\left(ky - th\right)\right), \cos \left(ky + th\right)\right), 1\right), \left(2 \cdot \sin ky\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \cos \left(ky + th\right)\right), 1\right), \left(2 \cdot \sin ky\right)\right) \]

      15. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \mathsf{cos.f64}\left(\left(ky + th\right)\right)\right), 1\right), \left(2 \cdot \sin ky\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(ky, th\right)\right)\right), 1\right), \left(2 \cdot \sin ky\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(ky, th\right)\right)\right), 1\right), \mathsf{*.f64}\left(2, \color{blue}{\sin ky}\right)\right) \]

      18. sin-lowering-sin.f64 20.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(ky, th\right)\right)\right), 1\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(ky\right)\right)\right) \]

   9. Applied egg-rr 20.0%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(ky - th\right) - \cos \left(ky + th\right)\right) \cdot 1}{2 \cdot \sin ky}} \]

   10. Taylor expanded in ky around 0

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\cos \left(\mathsf{neg}\left(th\right)\right) - \cos th}{ky}} \]
   11. Step-by-step derivation

       1. cos-neg N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\cos \left(\mathsf{neg}\left(th\right)\right) - \cos \left(\mathsf{neg}\left(th\right)\right)}{ky} \]

       2. div-sub N/A

          \[\leadsto \frac{1}{2} \cdot \left(\frac{\cos \left(\mathsf{neg}\left(th\right)\right)}{ky} - \color{blue}{\frac{\cos \left(\mathsf{neg}\left(th\right)\right)}{ky}}\right) \]

       3. +-inverses N/A

          \[\leadsto \frac{1}{2} \cdot 0 \]

       4. metadata-eval 19.8%

          \[\leadsto 0 \]

   12. Simplified 19.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 12.2% accurate, 709.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 0.0)

C

double code(double kx, double ky, double th) {
	return 0.0;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = 0.0d0
end function

Java

public static double code(double kx, double ky, double th) {
	return 0.0;
}

Python

def code(kx, ky, th):
	return 0.0

Julia

function code(kx, ky, th)
	return 0.0
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = 0.0;
end

Wolfram

code[kx_, ky_, th_] := 0.0

Derivation

1. Initial program 92.9%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   2. associate-/l* N/A

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin ky, \color{blue}{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \left(\frac{\color{blue}{\sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\sin th, \color{blue}{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}\right)\right) \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)\right)\right) \]

   10. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \left(\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right)\right) \]

   11. hypot-lowering-hypot.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\sin ky, \color{blue}{\sin kx}\right)\right)\right) \]

   12. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \sin \color{blue}{kx}\right)\right)\right) \]

   13. sin-lowering-sin.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(th\right), \mathsf{hypot.f64}\left(\mathsf{sin.f64}\left(ky\right), \mathsf{sin.f64}\left(kx\right)\right)\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
4. Add Preprocessing

5. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
6. Step-by-step derivation

   1. sin-lowering-sin.f64 21.7%

      \[\leadsto \mathsf{sin.f64}\left(th\right) \]

7. Simplified 21.7%

   \[\leadsto \color{blue}{\sin th} \]
8. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \sin th \cdot \color{blue}{1} \]

   2. lft-mult-inverse N/A

      \[\leadsto \sin th \cdot \left(\frac{1}{\sin ky} \cdot \color{blue}{\sin ky}\right) \]

   3. associate-*l* N/A

      \[\leadsto \left(\sin th \cdot \frac{1}{\sin ky}\right) \cdot \color{blue}{\sin ky} \]

   4. div-inv N/A

      \[\leadsto \frac{\sin th}{\sin ky} \cdot \sin \color{blue}{ky} \]

   5. *-commutative N/A

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sin ky}} \]

   7. div-inv N/A

      \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \color{blue}{\frac{1}{\sin ky}} \]

   8. sin-mult N/A

      \[\leadsto \frac{\cos \left(ky - th\right) - \cos \left(ky + th\right)}{2} \cdot \frac{\color{blue}{1}}{\sin ky} \]

   9. frac-times N/A

      \[\leadsto \frac{\left(\cos \left(ky - th\right) - \cos \left(ky + th\right)\right) \cdot 1}{\color{blue}{2 \cdot \sin ky}} \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(\cos \left(ky - th\right) - \cos \left(ky + th\right)\right) \cdot 1\right), \color{blue}{\left(2 \cdot \sin ky\right)}\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\cos \left(ky - th\right) - \cos \left(ky + th\right)\right), 1\right), \left(\color{blue}{2} \cdot \sin ky\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\cos \left(ky - th\right), \cos \left(ky + th\right)\right), 1\right), \left(2 \cdot \sin ky\right)\right) \]

   13. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\left(ky - th\right)\right), \cos \left(ky + th\right)\right), 1\right), \left(2 \cdot \sin ky\right)\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \cos \left(ky + th\right)\right), 1\right), \left(2 \cdot \sin ky\right)\right) \]

   15. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \mathsf{cos.f64}\left(\left(ky + th\right)\right)\right), 1\right), \left(2 \cdot \sin ky\right)\right) \]

   16. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(ky, th\right)\right)\right), 1\right), \left(2 \cdot \sin ky\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(ky, th\right)\right)\right), 1\right), \mathsf{*.f64}\left(2, \color{blue}{\sin ky}\right)\right) \]

   18. sin-lowering-sin.f64 12.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(ky, th\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(ky, th\right)\right)\right), 1\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(ky\right)\right)\right) \]

9. Applied egg-rr 12.6%

   \[\leadsto \color{blue}{\frac{\left(\cos \left(ky - th\right) - \cos \left(ky + th\right)\right) \cdot 1}{2 \cdot \sin ky}} \]

10. Taylor expanded in ky around 0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\cos \left(\mathsf{neg}\left(th\right)\right) - \cos th}{ky}} \]
11. Step-by-step derivation

    1. cos-neg N/A

       \[\leadsto \frac{1}{2} \cdot \frac{\cos \left(\mathsf{neg}\left(th\right)\right) - \cos \left(\mathsf{neg}\left(th\right)\right)}{ky} \]

    2. div-sub N/A

       \[\leadsto \frac{1}{2} \cdot \left(\frac{\cos \left(\mathsf{neg}\left(th\right)\right)}{ky} - \color{blue}{\frac{\cos \left(\mathsf{neg}\left(th\right)\right)}{ky}}\right) \]

    3. +-inverses N/A

       \[\leadsto \frac{1}{2} \cdot 0 \]

    4. metadata-eval 12.5%

       \[\leadsto 0 \]

12. Simplified 12.5%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(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)))