Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 19.9s

Alternatives: 15

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: 94.0% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

   1. +-commutative 94.8%

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

   2. unpow2 94.8%

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

   3. unpow2 94.8%

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

   4. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 0.0002:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.005)
   (- (sin th))
   (if (<= (sin ky) 0.0002)
     (* (sin th) (/ (sin ky) (hypot ky (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.005) {
		tmp = -sin(th);
	} else if (sin(ky) <= 0.0002) {
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.005) {
		tmp = -Math.sin(th);
	} else if (Math.sin(ky) <= 0.0002) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.005:
		tmp = -math.sin(th)
	elif math.sin(ky) <= 0.0002:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.005)
		tmp = Float64(-sin(th));
	elseif (sin(ky) <= 0.0002)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(ky, sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.005)
		tmp = -sin(th);
	elseif (sin(ky) <= 0.0002)
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.005], (-N[Sin[th], $MachinePrecision]), If[LessEqual[N[Sin[ky], $MachinePrecision], 0.0002], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0050000000000000001

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 7.4%

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

   5. Taylor expanded in ky around -inf 6.0%

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

      1. associate-*r/ 6.0%

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

      2. neg-mul-1 6.0%

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

      3. distribute-lft-neg-in 6.0%

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

      4. *-commutative 6.0%

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

   7. Simplified 6.0%

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

   8. Taylor expanded in ky around 0 57.9%

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

      1. mul-1-neg 57.9%

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

   10. Simplified 57.9%

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

   if -0.0050000000000000001 < (sin.f64 ky) < 2.0000000000000001e-4

   1. Initial program 89.4%

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

      1. +-commutative 89.4%

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

      2. unpow2 89.4%

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

      3. unpow2 89.4%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 99.5%

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

   if 2.0000000000000001e-4 < (sin.f64 ky)

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 64.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 0.0002:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 0.0002:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.005)
   (- (sin th))
   (if (<= (sin ky) 0.0002) (* (sin th) (/ ky (hypot ky (sin kx)))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.005) {
		tmp = -sin(th);
	} else if (sin(ky) <= 0.0002) {
		tmp = sin(th) * (ky / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.005) {
		tmp = -Math.sin(th);
	} else if (Math.sin(ky) <= 0.0002) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.005:
		tmp = -math.sin(th)
	elif math.sin(ky) <= 0.0002:
		tmp = math.sin(th) * (ky / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.005)
		tmp = Float64(-sin(th));
	elseif (sin(ky) <= 0.0002)
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.005)
		tmp = -sin(th);
	elseif (sin(ky) <= 0.0002)
		tmp = sin(th) * (ky / hypot(ky, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.005], (-N[Sin[th], $MachinePrecision]), If[LessEqual[N[Sin[ky], $MachinePrecision], 0.0002], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0050000000000000001

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 7.4%

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

   5. Taylor expanded in ky around -inf 6.0%

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

      1. associate-*r/ 6.0%

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

      2. neg-mul-1 6.0%

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

      3. distribute-lft-neg-in 6.0%

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

      4. *-commutative 6.0%

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

   7. Simplified 6.0%

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

   8. Taylor expanded in ky around 0 57.9%

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

      1. mul-1-neg 57.9%

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

   10. Simplified 57.9%

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

   if -0.0050000000000000001 < (sin.f64 ky) < 2.0000000000000001e-4

   1. Initial program 89.4%

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

      1. +-commutative 89.4%

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

      2. unpow2 89.4%

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

      3. unpow2 89.4%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 99.5%

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

   5. Taylor expanded in ky around 0 99.3%

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

   if 2.0000000000000001e-4 < (sin.f64 ky)

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 64.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 0.0002:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4: 68.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-36}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -5e-36)
   (- (sin th))
   (if (<= (sin ky) 2e-36) (* (sin th) (/ ky (fabs (sin kx)))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -5e-36) {
		tmp = -sin(th);
	} else if (sin(ky) <= 2e-36) {
		tmp = sin(th) * (ky / fabs(sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-5d-36)) then
        tmp = -sin(th)
    else if (sin(ky) <= 2d-36) then
        tmp = sin(th) * (ky / abs(sin(kx)))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -5e-36) {
		tmp = -Math.sin(th);
	} else if (Math.sin(ky) <= 2e-36) {
		tmp = Math.sin(th) * (ky / Math.abs(Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -5e-36:
		tmp = -math.sin(th)
	elif math.sin(ky) <= 2e-36:
		tmp = math.sin(th) * (ky / math.fabs(math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -5e-36)
		tmp = Float64(-sin(th));
	elseif (sin(ky) <= 2e-36)
		tmp = Float64(sin(th) * Float64(ky / abs(sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -5e-36)
		tmp = -sin(th);
	elseif (sin(ky) <= 2e-36)
		tmp = sin(th) * (ky / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-36], (-N[Sin[th], $MachinePrecision]), If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-36], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -5.00000000000000004e-36

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 11.1%

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

   5. Taylor expanded in ky around -inf 8.6%

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

      1. associate-*r/ 8.6%

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

      2. neg-mul-1 8.6%

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

      3. distribute-lft-neg-in 8.6%

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

      4. *-commutative 8.6%

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

   7. Simplified 8.6%

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

   8. Taylor expanded in ky around 0 58.4%

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

      1. mul-1-neg 58.4%

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

   10. Simplified 58.4%

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

   if -5.00000000000000004e-36 < (sin.f64 ky) < 1.9999999999999999e-36

   1. Initial program 88.6%

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

      1. +-commutative 88.6%

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

      2. unpow2 88.6%

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

      3. unpow2 88.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 47.2%

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

      1. add-sqr-sqrt 37.3%

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

      2. sqrt-prod 74.0%

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

      3. rem-sqrt-square 78.0%

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

   6. Applied egg-rr 78.0%

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

   if 1.9999999999999999e-36 < (sin.f64 ky)

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 65.2%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-36}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5: 68.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-36}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\frac{ky}{\frac{\left|\sin kx\right|}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -5e-36)
   (- (sin th))
   (if (<= (sin ky) 2e-36) (/ ky (/ (fabs (sin kx)) (sin th))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -5e-36) {
		tmp = -sin(th);
	} else if (sin(ky) <= 2e-36) {
		tmp = ky / (fabs(sin(kx)) / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-5d-36)) then
        tmp = -sin(th)
    else if (sin(ky) <= 2d-36) then
        tmp = ky / (abs(sin(kx)) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -5e-36) {
		tmp = -Math.sin(th);
	} else if (Math.sin(ky) <= 2e-36) {
		tmp = ky / (Math.abs(Math.sin(kx)) / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -5e-36:
		tmp = -math.sin(th)
	elif math.sin(ky) <= 2e-36:
		tmp = ky / (math.fabs(math.sin(kx)) / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -5e-36)
		tmp = Float64(-sin(th));
	elseif (sin(ky) <= 2e-36)
		tmp = Float64(ky / Float64(abs(sin(kx)) / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -5e-36)
		tmp = -sin(th);
	elseif (sin(ky) <= 2e-36)
		tmp = ky / (abs(sin(kx)) / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-36], (-N[Sin[th], $MachinePrecision]), If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-36], N[(ky / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -5.00000000000000004e-36

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 11.1%

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

   5. Taylor expanded in ky around -inf 8.6%

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

      1. associate-*r/ 8.6%

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

      2. neg-mul-1 8.6%

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

      3. distribute-lft-neg-in 8.6%

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

      4. *-commutative 8.6%

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

   7. Simplified 8.6%

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

   8. Taylor expanded in ky around 0 58.4%

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

      1. mul-1-neg 58.4%

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

   10. Simplified 58.4%

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

   if -5.00000000000000004e-36 < (sin.f64 ky) < 1.9999999999999999e-36

   1. Initial program 88.6%

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

   2. Taylor expanded in ky around 0 44.8%

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

      1. *-commutative 44.8%

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

      2. associate-/l* 47.2%

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

   4. Simplified 47.2%

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

      1. add-sqr-sqrt 37.3%

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

      2. sqrt-prod 74.0%

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

      3. rem-sqrt-square 78.0%

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

   6. Applied egg-rr 78.2%

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

   if 1.9999999999999999e-36 < (sin.f64 ky)

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 65.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-36}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\frac{ky}{\frac{\left|\sin kx\right|}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6: 80.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq -4 \cdot 10^{-5} \lor \neg \left(th \leq 3.7 \cdot 10^{-29}\right):\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= th -4e-5) (not (<= th 3.7e-29)))
   (* (sin th) (/ ky (hypot ky (sin kx))))
   (* th (/ (sin ky) (hypot (sin kx) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((th <= -4e-5) || !(th <= 3.7e-29)) {
		tmp = sin(th) * (ky / hypot(ky, sin(kx)));
	} else {
		tmp = th * (sin(ky) / hypot(sin(kx), sin(ky)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((th <= -4e-5) || !(th <= 3.7e-29)) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = th * (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (th <= -4e-5) or not (th <= 3.7e-29):
		tmp = math.sin(th) * (ky / math.hypot(ky, math.sin(kx)))
	else:
		tmp = th * (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((th <= -4e-5) || !(th <= 3.7e-29))
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, sin(kx))));
	else
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(kx), sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((th <= -4e-5) || ~((th <= 3.7e-29)))
		tmp = sin(th) * (ky / hypot(ky, sin(kx)));
	else
		tmp = th * (sin(ky) / hypot(sin(kx), sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[th, -4e-5], N[Not[LessEqual[th, 3.7e-29]], $MachinePrecision]], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < -4.00000000000000033e-5 or 3.6999999999999997e-29 < th

   1. Initial program 95.1%

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

      1. +-commutative 95.1%

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

      2. unpow2 95.1%

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

      3. unpow2 95.1%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 56.7%

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

   5. Taylor expanded in ky around 0 70.6%

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

   if -4.00000000000000033e-5 < th < 3.6999999999999997e-29

   1. Initial program 94.4%

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

      1. associate-/r/ 94.3%

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

      2. +-commutative 94.3%

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

      3. unpow2 94.3%

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

      4. sqr-neg 94.3%

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

      5. sin-neg 94.3%

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

      6. sin-neg 94.3%

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

      7. unpow2 94.3%

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

      8. +-commutative 94.3%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 94.2%

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

      1. associate-*r/ 94.3%

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

      2. +-commutative 94.3%

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

      3. unpow2 94.3%

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

      4. unpow2 94.3%

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

      5. hypot-def 99.7%

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

      6. *-rgt-identity 99.7%

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

      7. hypot-def 94.3%

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

      8. unpow2 94.3%

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

      9. unpow2 94.3%

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

      10. +-commutative 94.3%

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

      11. unpow2 94.3%

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

      12. unpow2 94.3%

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

      13. hypot-def 99.7%

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

   6. Simplified 99.7%

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

      1. associate-/r/ 99.8%

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

      2. hypot-udef 94.4%

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

      3. +-commutative 94.4%

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

      4. hypot-udef 99.8%

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

      5. add-sqr-sqrt 99.3%

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

      6. add-sqr-sqrt 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq -4 \cdot 10^{-5} \lor \neg \left(th \leq 3.7 \cdot 10^{-29}\right):\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \end{array} \]

Alternative 7: 40.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.75 \cdot 10^{-153}:\\ \;\;\;\;\frac{ky \cdot t_1}{ky}\\ \mathbf{elif}\;ky \leq 1.15 \cdot 10^{-38}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= ky -6.5e+222)
     t_1
     (if (<= ky -2.2e+136)
       (sin th)
       (if (<= ky -1.75e-153)
         (/ (* ky t_1) ky)
         (if (<= ky 1.15e-38) (* (sin th) (/ ky (sin kx))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (ky <= -6.5e+222) {
		tmp = t_1;
	} else if (ky <= -2.2e+136) {
		tmp = sin(th);
	} else if (ky <= -1.75e-153) {
		tmp = (ky * t_1) / ky;
	} else if (ky <= 1.15e-38) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (ky <= (-6.5d+222)) then
        tmp = t_1
    else if (ky <= (-2.2d+136)) then
        tmp = sin(th)
    else if (ky <= (-1.75d-153)) then
        tmp = (ky * t_1) / ky
    else if (ky <= 1.15d-38) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double tmp;
	if (ky <= -6.5e+222) {
		tmp = t_1;
	} else if (ky <= -2.2e+136) {
		tmp = Math.sin(th);
	} else if (ky <= -1.75e-153) {
		tmp = (ky * t_1) / ky;
	} else if (ky <= 1.15e-38) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if ky <= -6.5e+222:
		tmp = t_1
	elif ky <= -2.2e+136:
		tmp = math.sin(th)
	elif ky <= -1.75e-153:
		tmp = (ky * t_1) / ky
	elif ky <= 1.15e-38:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (ky <= -6.5e+222)
		tmp = t_1;
	elseif (ky <= -2.2e+136)
		tmp = sin(th);
	elseif (ky <= -1.75e-153)
		tmp = Float64(Float64(ky * t_1) / ky);
	elseif (ky <= 1.15e-38)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (ky <= -6.5e+222)
		tmp = t_1;
	elseif (ky <= -2.2e+136)
		tmp = sin(th);
	elseif (ky <= -1.75e-153)
		tmp = (ky * t_1) / ky;
	elseif (ky <= 1.15e-38)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[ky, -6.5e+222], t$95$1, If[LessEqual[ky, -2.2e+136], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -1.75e-153], N[(N[(ky * t$95$1), $MachinePrecision] / ky), $MachinePrecision], If[LessEqual[ky, 1.15e-38], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if ky < -6.5000000000000006e222

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 4.5%

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

   5. Taylor expanded in ky around -inf 4.5%

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

      1. associate-*r/ 4.5%

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

      2. neg-mul-1 4.5%

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

      3. distribute-lft-neg-in 4.5%

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

      4. *-commutative 4.5%

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

   7. Simplified 4.5%

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

   8. Taylor expanded in ky around 0 38.0%

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

      1. mul-1-neg 38.0%

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

   10. Simplified 38.0%

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

   if -6.5000000000000006e222 < ky < -2.1999999999999999e136 or 1.15000000000000001e-38 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 42.9%

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

   if -2.1999999999999999e136 < ky < -1.7499999999999999e-153

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 45.1%

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

   5. Taylor expanded in ky around -inf 25.2%

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

      1. associate-*r/ 25.2%

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

      2. neg-mul-1 25.2%

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

      3. distribute-lft-neg-in 25.2%

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

      4. *-commutative 25.2%

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

   7. Simplified 25.2%

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

   8. Taylor expanded in ky around 0 43.5%

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

      1. neg-mul-1 43.5%

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

      2. distribute-rgt-neg-in 43.5%

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

   10. Simplified 43.5%

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

   if -1.7499999999999999e-153 < ky < 1.15000000000000001e-38

   1. Initial program 85.7%

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

      1. +-commutative 85.7%

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

      2. unpow2 85.7%

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

      3. unpow2 85.7%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 54.2%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.75 \cdot 10^{-153}:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{ky}\\ \mathbf{elif}\;ky \leq 1.15 \cdot 10^{-38}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8: 40.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.2 \cdot 10^{-152}:\\ \;\;\;\;\frac{ky \cdot t_1}{ky}\\ \mathbf{elif}\;ky \leq 3.8 \cdot 10^{-38}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= ky -6.5e+222)
     t_1
     (if (<= ky -2.2e+136)
       (sin th)
       (if (<= ky -1.2e-152)
         (/ (* ky t_1) ky)
         (if (<= ky 3.8e-38) (* ky (/ (sin th) (sin kx))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (ky <= -6.5e+222) {
		tmp = t_1;
	} else if (ky <= -2.2e+136) {
		tmp = sin(th);
	} else if (ky <= -1.2e-152) {
		tmp = (ky * t_1) / ky;
	} else if (ky <= 3.8e-38) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (ky <= (-6.5d+222)) then
        tmp = t_1
    else if (ky <= (-2.2d+136)) then
        tmp = sin(th)
    else if (ky <= (-1.2d-152)) then
        tmp = (ky * t_1) / ky
    else if (ky <= 3.8d-38) then
        tmp = ky * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double tmp;
	if (ky <= -6.5e+222) {
		tmp = t_1;
	} else if (ky <= -2.2e+136) {
		tmp = Math.sin(th);
	} else if (ky <= -1.2e-152) {
		tmp = (ky * t_1) / ky;
	} else if (ky <= 3.8e-38) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if ky <= -6.5e+222:
		tmp = t_1
	elif ky <= -2.2e+136:
		tmp = math.sin(th)
	elif ky <= -1.2e-152:
		tmp = (ky * t_1) / ky
	elif ky <= 3.8e-38:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (ky <= -6.5e+222)
		tmp = t_1;
	elseif (ky <= -2.2e+136)
		tmp = sin(th);
	elseif (ky <= -1.2e-152)
		tmp = Float64(Float64(ky * t_1) / ky);
	elseif (ky <= 3.8e-38)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (ky <= -6.5e+222)
		tmp = t_1;
	elseif (ky <= -2.2e+136)
		tmp = sin(th);
	elseif (ky <= -1.2e-152)
		tmp = (ky * t_1) / ky;
	elseif (ky <= 3.8e-38)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[ky, -6.5e+222], t$95$1, If[LessEqual[ky, -2.2e+136], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -1.2e-152], N[(N[(ky * t$95$1), $MachinePrecision] / ky), $MachinePrecision], If[LessEqual[ky, 3.8e-38], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if ky < -6.5000000000000006e222

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 4.5%

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

   5. Taylor expanded in ky around -inf 4.5%

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

      1. associate-*r/ 4.5%

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

      2. neg-mul-1 4.5%

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

      3. distribute-lft-neg-in 4.5%

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

      4. *-commutative 4.5%

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

   7. Simplified 4.5%

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

   8. Taylor expanded in ky around 0 38.0%

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

      1. mul-1-neg 38.0%

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

   10. Simplified 38.0%

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

   if -6.5000000000000006e222 < ky < -2.1999999999999999e136 or 3.8e-38 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 42.9%

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

   if -2.1999999999999999e136 < ky < -1.2e-152

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 45.1%

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

   5. Taylor expanded in ky around -inf 25.2%

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

      1. associate-*r/ 25.2%

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

      2. neg-mul-1 25.2%

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

      3. distribute-lft-neg-in 25.2%

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

      4. *-commutative 25.2%

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

   7. Simplified 25.2%

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

   8. Taylor expanded in ky around 0 43.5%

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

      1. neg-mul-1 43.5%

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

      2. distribute-rgt-neg-in 43.5%

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

   10. Simplified 43.5%

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

   if -1.2e-152 < ky < 3.8e-38

   1. Initial program 85.7%

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

   2. Taylor expanded in ky around 0 51.2%

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

      1. *-commutative 51.2%

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

      2. associate-/l* 54.3%

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

   4. Simplified 54.3%

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

      1. clear-num 54.1%

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

      2. associate-/r/ 54.3%

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

      3. clear-num 54.2%

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

   6. Applied egg-rr 54.2%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.2 \cdot 10^{-152}:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{ky}\\ \mathbf{elif}\;ky \leq 3.8 \cdot 10^{-38}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9: 40.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -7.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{ky \cdot t_1}{ky}\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= ky -6.5e+222)
     t_1
     (if (<= ky -2.2e+136)
       (sin th)
       (if (<= ky -7.2e-153)
         (/ (* ky t_1) ky)
         (if (<= ky 7.5e-38) (/ ky (/ (sin kx) (sin th))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (ky <= -6.5e+222) {
		tmp = t_1;
	} else if (ky <= -2.2e+136) {
		tmp = sin(th);
	} else if (ky <= -7.2e-153) {
		tmp = (ky * t_1) / ky;
	} else if (ky <= 7.5e-38) {
		tmp = ky / (sin(kx) / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (ky <= (-6.5d+222)) then
        tmp = t_1
    else if (ky <= (-2.2d+136)) then
        tmp = sin(th)
    else if (ky <= (-7.2d-153)) then
        tmp = (ky * t_1) / ky
    else if (ky <= 7.5d-38) then
        tmp = ky / (sin(kx) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double tmp;
	if (ky <= -6.5e+222) {
		tmp = t_1;
	} else if (ky <= -2.2e+136) {
		tmp = Math.sin(th);
	} else if (ky <= -7.2e-153) {
		tmp = (ky * t_1) / ky;
	} else if (ky <= 7.5e-38) {
		tmp = ky / (Math.sin(kx) / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if ky <= -6.5e+222:
		tmp = t_1
	elif ky <= -2.2e+136:
		tmp = math.sin(th)
	elif ky <= -7.2e-153:
		tmp = (ky * t_1) / ky
	elif ky <= 7.5e-38:
		tmp = ky / (math.sin(kx) / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (ky <= -6.5e+222)
		tmp = t_1;
	elseif (ky <= -2.2e+136)
		tmp = sin(th);
	elseif (ky <= -7.2e-153)
		tmp = Float64(Float64(ky * t_1) / ky);
	elseif (ky <= 7.5e-38)
		tmp = Float64(ky / Float64(sin(kx) / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (ky <= -6.5e+222)
		tmp = t_1;
	elseif (ky <= -2.2e+136)
		tmp = sin(th);
	elseif (ky <= -7.2e-153)
		tmp = (ky * t_1) / ky;
	elseif (ky <= 7.5e-38)
		tmp = ky / (sin(kx) / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[ky, -6.5e+222], t$95$1, If[LessEqual[ky, -2.2e+136], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -7.2e-153], N[(N[(ky * t$95$1), $MachinePrecision] / ky), $MachinePrecision], If[LessEqual[ky, 7.5e-38], N[(ky / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if ky < -6.5000000000000006e222

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 4.5%

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

   5. Taylor expanded in ky around -inf 4.5%

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

      1. associate-*r/ 4.5%

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

      2. neg-mul-1 4.5%

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

      3. distribute-lft-neg-in 4.5%

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

      4. *-commutative 4.5%

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

   7. Simplified 4.5%

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

   8. Taylor expanded in ky around 0 38.0%

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

      1. mul-1-neg 38.0%

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

   10. Simplified 38.0%

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

   if -6.5000000000000006e222 < ky < -2.1999999999999999e136 or 7.5e-38 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 42.9%

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

   if -2.1999999999999999e136 < ky < -7.1999999999999995e-153

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 45.1%

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

   5. Taylor expanded in ky around -inf 25.2%

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

      1. associate-*r/ 25.2%

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

      2. neg-mul-1 25.2%

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

      3. distribute-lft-neg-in 25.2%

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

      4. *-commutative 25.2%

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

   7. Simplified 25.2%

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

   8. Taylor expanded in ky around 0 43.5%

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

      1. neg-mul-1 43.5%

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

      2. distribute-rgt-neg-in 43.5%

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

   10. Simplified 43.5%

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

   if -7.1999999999999995e-153 < ky < 7.5e-38

   1. Initial program 85.7%

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

   2. Taylor expanded in ky around 0 51.2%

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

      1. *-commutative 51.2%

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

      2. associate-/l* 54.3%

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

   4. Simplified 54.3%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -7.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{ky}\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 36.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -5.4 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 6.6 \cdot 10^{-97}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= ky -6.5e+222)
     t_1
     (if (<= ky -2.2e+136)
       (sin th)
       (if (<= ky -5.4e-180)
         t_1
         (if (<= ky 6.6e-97) (* (sin th) (/ ky kx)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (ky <= -6.5e+222) {
		tmp = t_1;
	} else if (ky <= -2.2e+136) {
		tmp = sin(th);
	} else if (ky <= -5.4e-180) {
		tmp = t_1;
	} else if (ky <= 6.6e-97) {
		tmp = sin(th) * (ky / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (ky <= (-6.5d+222)) then
        tmp = t_1
    else if (ky <= (-2.2d+136)) then
        tmp = sin(th)
    else if (ky <= (-5.4d-180)) then
        tmp = t_1
    else if (ky <= 6.6d-97) then
        tmp = sin(th) * (ky / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double tmp;
	if (ky <= -6.5e+222) {
		tmp = t_1;
	} else if (ky <= -2.2e+136) {
		tmp = Math.sin(th);
	} else if (ky <= -5.4e-180) {
		tmp = t_1;
	} else if (ky <= 6.6e-97) {
		tmp = Math.sin(th) * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if ky <= -6.5e+222:
		tmp = t_1
	elif ky <= -2.2e+136:
		tmp = math.sin(th)
	elif ky <= -5.4e-180:
		tmp = t_1
	elif ky <= 6.6e-97:
		tmp = math.sin(th) * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (ky <= -6.5e+222)
		tmp = t_1;
	elseif (ky <= -2.2e+136)
		tmp = sin(th);
	elseif (ky <= -5.4e-180)
		tmp = t_1;
	elseif (ky <= 6.6e-97)
		tmp = Float64(sin(th) * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (ky <= -6.5e+222)
		tmp = t_1;
	elseif (ky <= -2.2e+136)
		tmp = sin(th);
	elseif (ky <= -5.4e-180)
		tmp = t_1;
	elseif (ky <= 6.6e-97)
		tmp = sin(th) * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[ky, -6.5e+222], t$95$1, If[LessEqual[ky, -2.2e+136], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -5.4e-180], t$95$1, If[LessEqual[ky, 6.6e-97], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -6.5000000000000006e222 or -2.1999999999999999e136 < ky < -5.40000000000000028e-180

   1. Initial program 96.4%

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

      1. +-commutative 96.4%

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

      2. unpow2 96.4%

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

      3. unpow2 96.4%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 40.3%

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

   5. Taylor expanded in ky around -inf 22.2%

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

      1. associate-*r/ 22.2%

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

      2. neg-mul-1 22.2%

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

      3. distribute-lft-neg-in 22.2%

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

      4. *-commutative 22.2%

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

   7. Simplified 22.2%

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

   8. Taylor expanded in ky around 0 41.4%

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

      1. mul-1-neg 41.4%

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

   10. Simplified 41.4%

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

   if -6.5000000000000006e222 < ky < -2.1999999999999999e136 or 6.6000000000000002e-97 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 41.4%

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

   if -5.40000000000000028e-180 < ky < 6.6000000000000002e-97

   1. Initial program 85.3%

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

      1. +-commutative 85.3%

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

      2. unpow2 85.3%

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

      3. unpow2 85.3%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 55.8%

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

   5. Taylor expanded in kx around 0 39.4%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -5.4 \cdot 10^{-180}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq 6.6 \cdot 10^{-97}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 36.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -9 \cdot 10^{-179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 2.7 \cdot 10^{-96}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= ky -6.5e+222)
     t_1
     (if (<= ky -2.2e+136)
       (sin th)
       (if (<= ky -9e-179)
         t_1
         (if (<= ky 2.7e-96) (/ ky (/ kx (sin th))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (ky <= -6.5e+222) {
		tmp = t_1;
	} else if (ky <= -2.2e+136) {
		tmp = sin(th);
	} else if (ky <= -9e-179) {
		tmp = t_1;
	} else if (ky <= 2.7e-96) {
		tmp = ky / (kx / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (ky <= (-6.5d+222)) then
        tmp = t_1
    else if (ky <= (-2.2d+136)) then
        tmp = sin(th)
    else if (ky <= (-9d-179)) then
        tmp = t_1
    else if (ky <= 2.7d-96) then
        tmp = ky / (kx / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double tmp;
	if (ky <= -6.5e+222) {
		tmp = t_1;
	} else if (ky <= -2.2e+136) {
		tmp = Math.sin(th);
	} else if (ky <= -9e-179) {
		tmp = t_1;
	} else if (ky <= 2.7e-96) {
		tmp = ky / (kx / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if ky <= -6.5e+222:
		tmp = t_1
	elif ky <= -2.2e+136:
		tmp = math.sin(th)
	elif ky <= -9e-179:
		tmp = t_1
	elif ky <= 2.7e-96:
		tmp = ky / (kx / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (ky <= -6.5e+222)
		tmp = t_1;
	elseif (ky <= -2.2e+136)
		tmp = sin(th);
	elseif (ky <= -9e-179)
		tmp = t_1;
	elseif (ky <= 2.7e-96)
		tmp = Float64(ky / Float64(kx / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (ky <= -6.5e+222)
		tmp = t_1;
	elseif (ky <= -2.2e+136)
		tmp = sin(th);
	elseif (ky <= -9e-179)
		tmp = t_1;
	elseif (ky <= 2.7e-96)
		tmp = ky / (kx / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[ky, -6.5e+222], t$95$1, If[LessEqual[ky, -2.2e+136], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -9e-179], t$95$1, If[LessEqual[ky, 2.7e-96], N[(ky / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -6.5000000000000006e222 or -2.1999999999999999e136 < ky < -8.99999999999999984e-179

   1. Initial program 96.4%

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

      1. +-commutative 96.4%

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

      2. unpow2 96.4%

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

      3. unpow2 96.4%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 40.3%

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

   5. Taylor expanded in ky around -inf 22.2%

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

      1. associate-*r/ 22.2%

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

      2. neg-mul-1 22.2%

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

      3. distribute-lft-neg-in 22.2%

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

      4. *-commutative 22.2%

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

   7. Simplified 22.2%

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

   8. Taylor expanded in ky around 0 41.4%

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

      1. mul-1-neg 41.4%

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

   10. Simplified 41.4%

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

   if -6.5000000000000006e222 < ky < -2.1999999999999999e136 or 2.7e-96 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 41.4%

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

   if -8.99999999999999984e-179 < ky < 2.7e-96

   1. Initial program 85.3%

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

   2. Taylor expanded in ky around 0 51.5%

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

      1. *-commutative 51.5%

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

      2. associate-/l* 55.8%

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

   4. Simplified 55.8%

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

   5. Taylor expanded in kx around 0 39.4%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -9 \cdot 10^{-179}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq 2.7 \cdot 10^{-96}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 36.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.9 \cdot 10^{-197}:\\ \;\;\;\;\frac{ky \cdot t_1}{ky}\\ \mathbf{elif}\;ky \leq 7 \cdot 10^{-97}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= ky -6.5e+222)
     t_1
     (if (<= ky -2.2e+136)
       (sin th)
       (if (<= ky -1.9e-197)
         (/ (* ky t_1) ky)
         (if (<= ky 7e-97) (/ ky (/ kx (sin th))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (ky <= -6.5e+222) {
		tmp = t_1;
	} else if (ky <= -2.2e+136) {
		tmp = sin(th);
	} else if (ky <= -1.9e-197) {
		tmp = (ky * t_1) / ky;
	} else if (ky <= 7e-97) {
		tmp = ky / (kx / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (ky <= (-6.5d+222)) then
        tmp = t_1
    else if (ky <= (-2.2d+136)) then
        tmp = sin(th)
    else if (ky <= (-1.9d-197)) then
        tmp = (ky * t_1) / ky
    else if (ky <= 7d-97) then
        tmp = ky / (kx / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double tmp;
	if (ky <= -6.5e+222) {
		tmp = t_1;
	} else if (ky <= -2.2e+136) {
		tmp = Math.sin(th);
	} else if (ky <= -1.9e-197) {
		tmp = (ky * t_1) / ky;
	} else if (ky <= 7e-97) {
		tmp = ky / (kx / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if ky <= -6.5e+222:
		tmp = t_1
	elif ky <= -2.2e+136:
		tmp = math.sin(th)
	elif ky <= -1.9e-197:
		tmp = (ky * t_1) / ky
	elif ky <= 7e-97:
		tmp = ky / (kx / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (ky <= -6.5e+222)
		tmp = t_1;
	elseif (ky <= -2.2e+136)
		tmp = sin(th);
	elseif (ky <= -1.9e-197)
		tmp = Float64(Float64(ky * t_1) / ky);
	elseif (ky <= 7e-97)
		tmp = Float64(ky / Float64(kx / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (ky <= -6.5e+222)
		tmp = t_1;
	elseif (ky <= -2.2e+136)
		tmp = sin(th);
	elseif (ky <= -1.9e-197)
		tmp = (ky * t_1) / ky;
	elseif (ky <= 7e-97)
		tmp = ky / (kx / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[ky, -6.5e+222], t$95$1, If[LessEqual[ky, -2.2e+136], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -1.9e-197], N[(N[(ky * t$95$1), $MachinePrecision] / ky), $MachinePrecision], If[LessEqual[ky, 7e-97], N[(ky / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if ky < -6.5000000000000006e222

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 4.5%

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

   5. Taylor expanded in ky around -inf 4.5%

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

      1. associate-*r/ 4.5%

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

      2. neg-mul-1 4.5%

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

      3. distribute-lft-neg-in 4.5%

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

      4. *-commutative 4.5%

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

   7. Simplified 4.5%

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

   8. Taylor expanded in ky around 0 38.0%

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

      1. mul-1-neg 38.0%

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

   10. Simplified 38.0%

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

   if -6.5000000000000006e222 < ky < -2.1999999999999999e136 or 7.00000000000000038e-97 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 41.4%

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

   if -2.1999999999999999e136 < ky < -1.8999999999999999e-197

   1. Initial program 93.2%

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

      1. +-commutative 93.2%

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

      2. unpow2 93.2%

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

      3. unpow2 93.2%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 54.4%

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

   5. Taylor expanded in ky around -inf 29.1%

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

      1. associate-*r/ 29.1%

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

      2. neg-mul-1 29.1%

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

      3. distribute-lft-neg-in 29.1%

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

      4. *-commutative 29.1%

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

   7. Simplified 29.1%

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

   8. Taylor expanded in ky around 0 44.3%

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

      1. neg-mul-1 44.3%

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

      2. distribute-rgt-neg-in 44.3%

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

   10. Simplified 44.3%

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

   if -1.8999999999999999e-197 < ky < 7.00000000000000038e-97

   1. Initial program 87.1%

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

   2. Taylor expanded in ky around 0 53.1%

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

      1. *-commutative 53.1%

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

      2. associate-/l* 57.8%

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

   4. Simplified 57.8%

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

   5. Taylor expanded in kx around 0 41.4%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.9 \cdot 10^{-197}:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{ky}\\ \mathbf{elif}\;ky \leq 7 \cdot 10^{-97}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 29.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222} \lor \neg \left(ky \leq -2.2 \cdot 10^{+136}\right) \land ky \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -6.5e+222) (and (not (<= ky -2.2e+136)) (<= ky -2e-310)))
   (- (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -6.5e+222) || (!(ky <= -2.2e+136) && (ky <= -2e-310))) {
		tmp = -sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((ky <= (-6.5d+222)) .or. (.not. (ky <= (-2.2d+136))) .and. (ky <= (-2d-310))) then
        tmp = -sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -6.5e+222) || (!(ky <= -2.2e+136) && (ky <= -2e-310))) {
		tmp = -Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -6.5e+222) or (not (ky <= -2.2e+136) and (ky <= -2e-310)):
		tmp = -math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -6.5e+222) || (!(ky <= -2.2e+136) && (ky <= -2e-310)))
		tmp = Float64(-sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -6.5e+222) || (~((ky <= -2.2e+136)) && (ky <= -2e-310)))
		tmp = -sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -6.5e+222], And[N[Not[LessEqual[ky, -2.2e+136]], $MachinePrecision], LessEqual[ky, -2e-310]]], (-N[Sin[th], $MachinePrecision]), N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -6.5000000000000006e222 or -2.1999999999999999e136 < ky < -1.999999999999994e-310

   1. Initial program 92.1%

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

      1. +-commutative 92.1%

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

      2. unpow2 92.1%

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

      3. unpow2 92.1%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 56.9%

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

   5. Taylor expanded in ky around -inf 27.3%

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

      1. associate-*r/ 27.3%

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

      2. neg-mul-1 27.3%

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

      3. distribute-lft-neg-in 27.3%

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

      4. *-commutative 27.3%

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

   7. Simplified 27.3%

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

   8. Taylor expanded in ky around 0 35.1%

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

      1. mul-1-neg 35.1%

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

   10. Simplified 35.1%

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

   if -6.5000000000000006e222 < ky < -2.1999999999999999e136 or -1.999999999999994e-310 < ky

   1. Initial program 97.4%

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

   2. Taylor expanded in kx around 0 35.8%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -6.5 \cdot 10^{+222} \lor \neg \left(ky \leq -2.2 \cdot 10^{+136}\right) \land ky \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 23.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \sin th \end{array} \]

FPCore

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

C

double code(double kx, double ky, double th) {
	return 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(th)
end function

Java

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

Python

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

Julia

function code(kx, ky, th)
	return sin(th)
end

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[Sin[th], $MachinePrecision]

Derivation

1. Initial program 94.8%

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

2. Taylor expanded in kx around 0 23.8%

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

3. Final simplification 23.8%

   \[\leadsto \sin th \]

Alternative 15: 13.6% accurate, 709.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

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

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 94.8%

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

   1. associate-/r/ 94.8%

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

   2. +-commutative 94.8%

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

   3. unpow2 94.8%

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

   4. sqr-neg 94.8%

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

   5. sin-neg 94.8%

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

   6. sin-neg 94.8%

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

   7. unpow2 94.8%

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

   8. +-commutative 94.8%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 44.0%

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

   1. associate-*r/ 44.1%

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

   2. +-commutative 44.1%

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

   3. unpow2 44.1%

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

   4. unpow2 44.1%

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

   5. hypot-def 46.4%

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

   6. *-rgt-identity 46.4%

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

   7. hypot-def 44.1%

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

   8. unpow2 44.1%

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

   9. unpow2 44.1%

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

   10. +-commutative 44.1%

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

   11. unpow2 44.1%

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

   12. unpow2 44.1%

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

   13. hypot-def 46.4%

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

6. Simplified 46.4%

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

7. Taylor expanded in kx around 0 10.5%

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

8. Final simplification 10.5%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023264 
(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)))