Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.5% → 99.6%

Time: 15.5s

Alternatives: 16

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. unpow2 94.0%

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

   2. sqr-neg 94.0%

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

   3. sin-neg 94.0%

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

   4. sin-neg 94.0%

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

   5. unpow2 94.0%

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

   6. associate-*l/ 91.9%

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

   7. associate-/l* 94.0%

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

   8. +-commutative 94.0%

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

   9. unpow2 94.0%

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

   10. sin-neg 94.0%

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

   11. sin-neg 94.0%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.54:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq -0.005:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{elif}\;\sin th \leq 0.01:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin th \leq 0.78:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.54)
   (sin th)
   (if (<= (sin th) -0.005)
     (/ (sin th) (/ (sin kx) (sin ky)))
     (if (<= (sin th) 0.01)
       (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
       (if (<= (sin th) 0.78)
         (/ (sin ky) (fabs (/ (sin kx) (sin th))))
         (* (sin ky) (fabs (/ (sin th) (sin ky)))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.54) {
		tmp = sin(th);
	} else if (sin(th) <= -0.005) {
		tmp = sin(th) / (sin(kx) / sin(ky));
	} else if (sin(th) <= 0.01) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else if (sin(th) <= 0.78) {
		tmp = sin(ky) / fabs((sin(kx) / sin(th)));
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.54) {
		tmp = Math.sin(th);
	} else if (Math.sin(th) <= -0.005) {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	} else if (Math.sin(th) <= 0.01) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else if (Math.sin(th) <= 0.78) {
		tmp = Math.sin(ky) / Math.abs((Math.sin(kx) / Math.sin(th)));
	} else {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.54:
		tmp = math.sin(th)
	elif math.sin(th) <= -0.005:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	elif math.sin(th) <= 0.01:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	elif math.sin(th) <= 0.78:
		tmp = math.sin(ky) / math.fabs((math.sin(kx) / math.sin(th)))
	else:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.54)
		tmp = sin(th);
	elseif (sin(th) <= -0.005)
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	elseif (sin(th) <= 0.01)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	elseif (sin(th) <= 0.78)
		tmp = Float64(sin(ky) / abs(Float64(sin(kx) / sin(th))));
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.54)
		tmp = sin(th);
	elseif (sin(th) <= -0.005)
		tmp = sin(th) / (sin(kx) / sin(ky));
	elseif (sin(th) <= 0.01)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	elseif (sin(th) <= 0.78)
		tmp = sin(ky) / abs((sin(kx) / sin(th)));
	else
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.54], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], -0.005], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.01], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.78], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 th) < -0.54000000000000004

   1. Initial program 90.0%

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

      1. unpow2 90.0%

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

      2. sqr-neg 90.0%

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

      3. sin-neg 90.0%

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

      4. sin-neg 90.0%

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

      5. unpow2 90.0%

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

      6. associate-*l/ 90.0%

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

      7. associate-/l* 89.9%

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

      8. +-commutative 89.9%

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

      9. unpow2 89.9%

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

      10. sin-neg 89.9%

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

      11. sin-neg 89.9%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 18.5%

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

   if -0.54000000000000004 < (sin.f64 th) < -0.0050000000000000001

   1. Initial program 96.1%

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

      1. +-commutative 96.1%

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

      2. unpow2 96.1%

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

      3. unpow2 96.1%

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

      4. hypot-undefine 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. clear-num 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-un-lft-identity 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in ky around 0 27.3%

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

   if -0.0050000000000000001 < (sin.f64 th) < 0.0100000000000000002

   1. Initial program 94.5%

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

      1. unpow2 94.5%

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

      2. sqr-neg 94.5%

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

      3. sin-neg 94.5%

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

      4. sin-neg 94.5%

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

      5. unpow2 94.5%

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

      6. associate-*l/ 90.6%

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

      7. associate-/l* 94.5%

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

      8. +-commutative 94.5%

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

      9. unpow2 94.5%

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

      10. sin-neg 94.5%

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

      11. sin-neg 94.5%

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

   3. Simplified 99.7%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in th around 0 92.0%

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

      1. associate-*l/ 92.7%

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

      2. +-commutative 92.7%

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

      3. unpow2 92.7%

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

      4. unpow2 92.7%

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

      5. hypot-undefine 97.8%

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

      6. *-lft-identity 97.8%

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

      7. hypot-undefine 92.7%

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

      8. unpow2 92.7%

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

      9. unpow2 92.7%

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

      10. +-commutative 92.7%

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

      11. unpow2 92.7%

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

      12. unpow2 92.7%

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

      13. hypot-define 97.8%

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

   9. Simplified 97.8%

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

   if 0.0100000000000000002 < (sin.f64 th) < 0.78000000000000003

   1. Initial program 93.1%

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

      1. unpow2 93.1%

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

      2. sqr-neg 93.1%

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

      3. sin-neg 93.1%

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

      4. sin-neg 93.1%

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

      5. unpow2 93.1%

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

      6. associate-*l/ 92.9%

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

      7. associate-/l* 93.1%

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

      8. +-commutative 93.1%

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

      9. unpow2 93.1%

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

      10. sin-neg 93.1%

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

      11. sin-neg 93.1%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in ky around 0 30.7%

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

      1. add-sqr-sqrt 29.3%

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

      2. sqrt-unprod 53.8%

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

      3. pow2 53.8%

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

   9. Applied egg-rr 53.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\left(\frac{\sin kx}{\sin th}\right)}^{2}}}} \]
   10. Step-by-step derivation

       1. unpow2 53.8%

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

       2. rem-sqrt-square 57.3%

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

   11. Simplified 57.3%

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

   if 0.78000000000000003 < (sin.f64 th)

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      6. associate-*l/ 99.2%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 15.0%

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

      1. add-sqr-sqrt 13.5%

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

      2. sqrt-unprod 38.9%

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

      3. pow2 38.9%

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

   7. Applied egg-rr 38.9%

      \[\leadsto \sin ky \cdot \color{blue}{\sqrt{{\left(\frac{\sin th}{\sin ky}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 38.9%

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

      2. rem-sqrt-square 39.5%

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

   9. Simplified 39.5%

      \[\leadsto \sin ky \cdot \color{blue}{\left|\frac{\sin th}{\sin ky}\right|} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.54:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq -0.005:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{elif}\;\sin th \leq 0.01:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin th \leq 0.78:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\sin ky \cdot \left|\frac{th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-110}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-74} \lor \neg \left(\sin ky \leq 10^{-50}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{\sin kx}{ky \cdot \sin th}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.004)
   (* (sin ky) (fabs (/ th (sin ky))))
   (if (<= (sin ky) 1e-110)
     (* ky (/ (sin th) (sin kx)))
     (if (or (<= (sin ky) 2e-74) (not (<= (sin ky) 1e-50)))
       (sin th)
       (/ 1.0 (fabs (/ (sin kx) (* ky (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.004) {
		tmp = sin(ky) * fabs((th / sin(ky)));
	} else if (sin(ky) <= 1e-110) {
		tmp = ky * (sin(th) / sin(kx));
	} else if ((sin(ky) <= 2e-74) || !(sin(ky) <= 1e-50)) {
		tmp = sin(th);
	} else {
		tmp = 1.0 / fabs((sin(kx) / (ky * sin(th))));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.004d0)) then
        tmp = sin(ky) * abs((th / sin(ky)))
    else if (sin(ky) <= 1d-110) then
        tmp = ky * (sin(th) / sin(kx))
    else if ((sin(ky) <= 2d-74) .or. (.not. (sin(ky) <= 1d-50))) then
        tmp = sin(th)
    else
        tmp = 1.0d0 / abs((sin(kx) / (ky * sin(th))))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.004) {
		tmp = Math.sin(ky) * Math.abs((th / Math.sin(ky)));
	} else if (Math.sin(ky) <= 1e-110) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else if ((Math.sin(ky) <= 2e-74) || !(Math.sin(ky) <= 1e-50)) {
		tmp = Math.sin(th);
	} else {
		tmp = 1.0 / Math.abs((Math.sin(kx) / (ky * Math.sin(th))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.004:
		tmp = math.sin(ky) * math.fabs((th / math.sin(ky)))
	elif math.sin(ky) <= 1e-110:
		tmp = ky * (math.sin(th) / math.sin(kx))
	elif (math.sin(ky) <= 2e-74) or not (math.sin(ky) <= 1e-50):
		tmp = math.sin(th)
	else:
		tmp = 1.0 / math.fabs((math.sin(kx) / (ky * math.sin(th))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.004)
		tmp = Float64(sin(ky) * abs(Float64(th / sin(ky))));
	elseif (sin(ky) <= 1e-110)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	elseif ((sin(ky) <= 2e-74) || !(sin(ky) <= 1e-50))
		tmp = sin(th);
	else
		tmp = Float64(1.0 / abs(Float64(sin(kx) / Float64(ky * sin(th)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.004)
		tmp = sin(ky) * abs((th / sin(ky)));
	elseif (sin(ky) <= 1e-110)
		tmp = ky * (sin(th) / sin(kx));
	elseif ((sin(ky) <= 2e-74) || ~((sin(ky) <= 1e-50)))
		tmp = sin(th);
	else
		tmp = 1.0 / abs((sin(kx) / (ky * sin(th))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.004], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(th / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-110], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[Sin[ky], $MachinePrecision], 2e-74], N[Not[LessEqual[N[Sin[ky], $MachinePrecision], 1e-50]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(1.0 / N[Abs[N[(N[Sin[kx], $MachinePrecision] / N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.0040000000000000001

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. +-commutative 99.7%

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

      9. unpow2 99.7%

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

      10. sin-neg 99.7%

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

      11. sin-neg 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.6%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 23.2%

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

      3. pow2 23.2%

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

   7. Applied egg-rr 23.2%

      \[\leadsto \sin ky \cdot \color{blue}{\sqrt{{\left(\frac{\sin th}{\sin ky}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 23.2%

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

      2. rem-sqrt-square 35.9%

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

   9. Simplified 35.9%

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

   10. Taylor expanded in th around 0 22.7%

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

   if -0.0040000000000000001 < (sin.f64 ky) < 1.0000000000000001e-110

   1. Initial program 86.6%

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

      1. unpow2 86.6%

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

      2. sqr-neg 86.6%

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

      3. sin-neg 86.6%

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

      4. sin-neg 86.6%

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

      5. unpow2 86.6%

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

      6. associate-*l/ 81.8%

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

      7. associate-/l* 86.6%

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

      8. +-commutative 86.6%

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

      9. unpow2 86.6%

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

      10. sin-neg 86.6%

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

      11. sin-neg 86.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 52.8%

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

      1. associate-/l* 54.5%

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

   7. Simplified 54.5%

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

   if 1.0000000000000001e-110 < (sin.f64 ky) < 1.99999999999999992e-74 or 1.00000000000000001e-50 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. +-commutative 99.7%

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

      9. unpow2 99.7%

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

      10. sin-neg 99.7%

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

      11. sin-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 60.0%

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

   if 1.99999999999999992e-74 < (sin.f64 ky) < 1.00000000000000001e-50

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.1%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.1%

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

      2. clear-num 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in ky around 0 44.5%

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

      1. add-sqr-sqrt 28.9%

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

      2. sqrt-unprod 45.4%

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

      3. pow2 45.4%

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

      4. *-commutative 45.4%

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

   9. Applied egg-rr 45.4%

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

       1. unpow2 45.4%

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

       2. rem-sqrt-square 58.2%

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

       3. *-commutative 58.2%

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

   11. Simplified 58.2%

       \[\leadsto \frac{1}{\color{blue}{\left|\frac{\sin kx}{ky \cdot \sin th}\right|}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\sin ky \cdot \left|\frac{th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-110}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-74} \lor \neg \left(\sin ky \leq 10^{-50}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{\sin kx}{ky \cdot \sin th}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-110}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-74} \lor \neg \left(\sin ky \leq 10^{-50}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{\sin kx}{ky \cdot \sin th}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.004)
   (* (sin ky) (fabs (/ (sin th) (sin ky))))
   (if (<= (sin ky) 1e-110)
     (* ky (/ (sin th) (sin kx)))
     (if (or (<= (sin ky) 2e-74) (not (<= (sin ky) 1e-50)))
       (sin th)
       (/ 1.0 (fabs (/ (sin kx) (* ky (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.004) {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	} else if (sin(ky) <= 1e-110) {
		tmp = ky * (sin(th) / sin(kx));
	} else if ((sin(ky) <= 2e-74) || !(sin(ky) <= 1e-50)) {
		tmp = sin(th);
	} else {
		tmp = 1.0 / fabs((sin(kx) / (ky * sin(th))));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.004d0)) then
        tmp = sin(ky) * abs((sin(th) / sin(ky)))
    else if (sin(ky) <= 1d-110) then
        tmp = ky * (sin(th) / sin(kx))
    else if ((sin(ky) <= 2d-74) .or. (.not. (sin(ky) <= 1d-50))) then
        tmp = sin(th)
    else
        tmp = 1.0d0 / abs((sin(kx) / (ky * sin(th))))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.004) {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	} else if (Math.sin(ky) <= 1e-110) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else if ((Math.sin(ky) <= 2e-74) || !(Math.sin(ky) <= 1e-50)) {
		tmp = Math.sin(th);
	} else {
		tmp = 1.0 / Math.abs((Math.sin(kx) / (ky * Math.sin(th))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.004:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	elif math.sin(ky) <= 1e-110:
		tmp = ky * (math.sin(th) / math.sin(kx))
	elif (math.sin(ky) <= 2e-74) or not (math.sin(ky) <= 1e-50):
		tmp = math.sin(th)
	else:
		tmp = 1.0 / math.fabs((math.sin(kx) / (ky * math.sin(th))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.004)
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	elseif (sin(ky) <= 1e-110)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	elseif ((sin(ky) <= 2e-74) || !(sin(ky) <= 1e-50))
		tmp = sin(th);
	else
		tmp = Float64(1.0 / abs(Float64(sin(kx) / Float64(ky * sin(th)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.004)
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	elseif (sin(ky) <= 1e-110)
		tmp = ky * (sin(th) / sin(kx));
	elseif ((sin(ky) <= 2e-74) || ~((sin(ky) <= 1e-50)))
		tmp = sin(th);
	else
		tmp = 1.0 / abs((sin(kx) / (ky * sin(th))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.004], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-110], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[Sin[ky], $MachinePrecision], 2e-74], N[Not[LessEqual[N[Sin[ky], $MachinePrecision], 1e-50]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(1.0 / N[Abs[N[(N[Sin[kx], $MachinePrecision] / N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.0040000000000000001

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. +-commutative 99.7%

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

      9. unpow2 99.7%

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

      10. sin-neg 99.7%

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

      11. sin-neg 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.6%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 23.2%

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

      3. pow2 23.2%

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

   7. Applied egg-rr 23.2%

      \[\leadsto \sin ky \cdot \color{blue}{\sqrt{{\left(\frac{\sin th}{\sin ky}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 23.2%

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

      2. rem-sqrt-square 35.9%

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

   9. Simplified 35.9%

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

   if -0.0040000000000000001 < (sin.f64 ky) < 1.0000000000000001e-110

   1. Initial program 86.6%

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

      1. unpow2 86.6%

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

      2. sqr-neg 86.6%

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

      3. sin-neg 86.6%

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

      4. sin-neg 86.6%

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

      5. unpow2 86.6%

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

      6. associate-*l/ 81.8%

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

      7. associate-/l* 86.6%

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

      8. +-commutative 86.6%

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

      9. unpow2 86.6%

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

      10. sin-neg 86.6%

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

      11. sin-neg 86.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 52.8%

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

      1. associate-/l* 54.5%

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

   7. Simplified 54.5%

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

   if 1.0000000000000001e-110 < (sin.f64 ky) < 1.99999999999999992e-74 or 1.00000000000000001e-50 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. +-commutative 99.7%

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

      9. unpow2 99.7%

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

      10. sin-neg 99.7%

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

      11. sin-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 60.0%

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

   if 1.99999999999999992e-74 < (sin.f64 ky) < 1.00000000000000001e-50

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.1%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.1%

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

      2. clear-num 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in ky around 0 44.5%

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

      1. add-sqr-sqrt 28.9%

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

      2. sqrt-unprod 45.4%

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

      3. pow2 45.4%

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

      4. *-commutative 45.4%

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

   9. Applied egg-rr 45.4%

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

       1. unpow2 45.4%

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

       2. rem-sqrt-square 58.2%

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

       3. *-commutative 58.2%

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

   11. Simplified 58.2%

       \[\leadsto \frac{1}{\color{blue}{\left|\frac{\sin kx}{ky \cdot \sin th}\right|}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-110}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-74} \lor \neg \left(\sin ky \leq 10^{-50}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{\sin kx}{ky \cdot \sin th}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\left|1 + \left(-1 - \sin th\right)\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-110} \lor \neg \left(\sin ky \leq 10^{-71}\right) \land \sin ky \leq 10^{-50}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.004)
   (fabs (+ 1.0 (- -1.0 (sin th))))
   (if (or (<= (sin ky) 1e-110)
           (and (not (<= (sin ky) 1e-71)) (<= (sin ky) 1e-50)))
     (* ky (/ (sin th) (sin kx)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.004) {
		tmp = fabs((1.0 + (-1.0 - sin(th))));
	} else if ((sin(ky) <= 1e-110) || (!(sin(ky) <= 1e-71) && (sin(ky) <= 1e-50))) {
		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) :: tmp
    if (sin(ky) <= (-0.004d0)) then
        tmp = abs((1.0d0 + ((-1.0d0) - sin(th))))
    else if ((sin(ky) <= 1d-110) .or. (.not. (sin(ky) <= 1d-71)) .and. (sin(ky) <= 1d-50)) 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 tmp;
	if (Math.sin(ky) <= -0.004) {
		tmp = Math.abs((1.0 + (-1.0 - Math.sin(th))));
	} else if ((Math.sin(ky) <= 1e-110) || (!(Math.sin(ky) <= 1e-71) && (Math.sin(ky) <= 1e-50))) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.004:
		tmp = math.fabs((1.0 + (-1.0 - math.sin(th))))
	elif (math.sin(ky) <= 1e-110) or (not (math.sin(ky) <= 1e-71) and (math.sin(ky) <= 1e-50)):
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.004)
		tmp = abs(Float64(1.0 + Float64(-1.0 - sin(th))));
	elseif ((sin(ky) <= 1e-110) || (!(sin(ky) <= 1e-71) && (sin(ky) <= 1e-50)))
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.004)
		tmp = abs((1.0 + (-1.0 - sin(th))));
	elseif ((sin(ky) <= 1e-110) || (~((sin(ky) <= 1e-71)) && (sin(ky) <= 1e-50)))
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.004], N[Abs[N[(1.0 + N[(-1.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[N[Sin[ky], $MachinePrecision], 1e-110], And[N[Not[LessEqual[N[Sin[ky], $MachinePrecision], 1e-71]], $MachinePrecision], LessEqual[N[Sin[ky], $MachinePrecision], 1e-50]]], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0040000000000000001

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. +-commutative 99.7%

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

      9. unpow2 99.7%

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

      10. sin-neg 99.7%

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

      11. sin-neg 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.6%

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

      1. expm1-log1p-u 2.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. expm1-undefine 3.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]

      3. associate-*r/ 3.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sin ky \cdot \sin th}{\sin ky}}\right)} - 1 \]

      4. *-commutative 3.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sin th \cdot \sin ky}}{\sin ky}\right)} - 1 \]

      5. associate-/l* 3.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sin th \cdot \frac{\sin ky}{\sin ky}}\right)} - 1 \]

   7. Applied egg-rr 3.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th \cdot \frac{\sin ky}{\sin ky}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-define 2.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th \cdot \frac{\sin ky}{\sin ky}\right)\right)} \]

      2. *-inverses 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin th \cdot \color{blue}{1}\right)\right) \]

      3. *-rgt-identity 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 2.6%

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

       2. add-sqr-sqrt 1.5%

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

       3. fabs-sqr 1.5%

          \[\leadsto \color{blue}{\left|\sqrt{\sin th} \cdot \sqrt{\sin th}\right|} \]

       4. add-sqr-sqrt 34.2%

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

       5. expm1-log1p-u 34.2%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)}\right| \]

       6. expm1-undefine 23.7%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1}\right| \]

       7. fabs-sub 23.7%

          \[\leadsto \color{blue}{\left|1 - e^{\mathsf{log1p}\left(\sin th\right)}\right|} \]

       8. log1p-undefine 23.7%

          \[\leadsto \left|1 - e^{\color{blue}{\log \left(1 + \sin th\right)}}\right| \]

       9. rem-exp-log 23.7%

          \[\leadsto \left|1 - \color{blue}{\left(1 + \sin th\right)}\right| \]

       10. +-commutative 23.7%

           \[\leadsto \left|1 - \color{blue}{\left(\sin th + 1\right)}\right| \]

   11. Applied egg-rr 23.7%

       \[\leadsto \color{blue}{\left|1 - \left(\sin th + 1\right)\right|} \]

   if -0.0040000000000000001 < (sin.f64 ky) < 1.0000000000000001e-110 or 9.9999999999999992e-72 < (sin.f64 ky) < 1.00000000000000001e-50

   1. Initial program 87.4%

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

      1. unpow2 87.4%

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

      2. sqr-neg 87.4%

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

      3. sin-neg 87.4%

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

      4. sin-neg 87.4%

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

      5. unpow2 87.4%

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

      6. associate-*l/ 82.9%

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

      7. associate-/l* 87.4%

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

      8. +-commutative 87.4%

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

      9. unpow2 87.4%

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

      10. sin-neg 87.4%

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

      11. sin-neg 87.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 52.3%

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

      1. associate-/l* 53.9%

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

   7. Simplified 53.9%

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

   if 1.0000000000000001e-110 < (sin.f64 ky) < 9.9999999999999992e-72 or 1.00000000000000001e-50 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. +-commutative 99.7%

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

      9. unpow2 99.7%

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

      10. sin-neg 99.7%

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

      11. sin-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 60.0%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\left|1 + \left(-1 - \sin th\right)\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-110} \lor \neg \left(\sin ky \leq 10^{-71}\right) \land \sin ky \leq 10^{-50}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\left|1 + \left(-1 - \sin th\right)\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-110}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-71} \lor \neg \left(\sin ky \leq 10^{-50}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.004)
   (fabs (+ 1.0 (- -1.0 (sin th))))
   (if (<= (sin ky) 1e-110)
     (* ky (/ (sin th) (sin kx)))
     (if (or (<= (sin ky) 1e-71) (not (<= (sin ky) 1e-50)))
       (sin th)
       (* (sin th) (/ ky (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.004) {
		tmp = fabs((1.0 + (-1.0 - sin(th))));
	} else if (sin(ky) <= 1e-110) {
		tmp = ky * (sin(th) / sin(kx));
	} else if ((sin(ky) <= 1e-71) || !(sin(ky) <= 1e-50)) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (ky / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.004d0)) then
        tmp = abs((1.0d0 + ((-1.0d0) - sin(th))))
    else if (sin(ky) <= 1d-110) then
        tmp = ky * (sin(th) / sin(kx))
    else if ((sin(ky) <= 1d-71) .or. (.not. (sin(ky) <= 1d-50))) then
        tmp = sin(th)
    else
        tmp = sin(th) * (ky / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.004) {
		tmp = Math.abs((1.0 + (-1.0 - Math.sin(th))));
	} else if (Math.sin(ky) <= 1e-110) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else if ((Math.sin(ky) <= 1e-71) || !(Math.sin(ky) <= 1e-50)) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.004:
		tmp = math.fabs((1.0 + (-1.0 - math.sin(th))))
	elif math.sin(ky) <= 1e-110:
		tmp = ky * (math.sin(th) / math.sin(kx))
	elif (math.sin(ky) <= 1e-71) or not (math.sin(ky) <= 1e-50):
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.004)
		tmp = abs(Float64(1.0 + Float64(-1.0 - sin(th))));
	elseif (sin(ky) <= 1e-110)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	elseif ((sin(ky) <= 1e-71) || !(sin(ky) <= 1e-50))
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.004)
		tmp = abs((1.0 + (-1.0 - sin(th))));
	elseif (sin(ky) <= 1e-110)
		tmp = ky * (sin(th) / sin(kx));
	elseif ((sin(ky) <= 1e-71) || ~((sin(ky) <= 1e-50)))
		tmp = sin(th);
	else
		tmp = sin(th) * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.004], N[Abs[N[(1.0 + N[(-1.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-110], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[Sin[ky], $MachinePrecision], 1e-71], N[Not[LessEqual[N[Sin[ky], $MachinePrecision], 1e-50]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.0040000000000000001

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. +-commutative 99.7%

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

      9. unpow2 99.7%

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

      10. sin-neg 99.7%

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

      11. sin-neg 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.6%

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

      1. expm1-log1p-u 2.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. expm1-undefine 3.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]

      3. associate-*r/ 3.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sin ky \cdot \sin th}{\sin ky}}\right)} - 1 \]

      4. *-commutative 3.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sin th \cdot \sin ky}}{\sin ky}\right)} - 1 \]

      5. associate-/l* 3.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sin th \cdot \frac{\sin ky}{\sin ky}}\right)} - 1 \]

   7. Applied egg-rr 3.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th \cdot \frac{\sin ky}{\sin ky}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-define 2.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th \cdot \frac{\sin ky}{\sin ky}\right)\right)} \]

      2. *-inverses 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin th \cdot \color{blue}{1}\right)\right) \]

      3. *-rgt-identity 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 2.6%

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

       2. add-sqr-sqrt 1.5%

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

       3. fabs-sqr 1.5%

          \[\leadsto \color{blue}{\left|\sqrt{\sin th} \cdot \sqrt{\sin th}\right|} \]

       4. add-sqr-sqrt 34.2%

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

       5. expm1-log1p-u 34.2%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)}\right| \]

       6. expm1-undefine 23.7%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1}\right| \]

       7. fabs-sub 23.7%

          \[\leadsto \color{blue}{\left|1 - e^{\mathsf{log1p}\left(\sin th\right)}\right|} \]

       8. log1p-undefine 23.7%

          \[\leadsto \left|1 - e^{\color{blue}{\log \left(1 + \sin th\right)}}\right| \]

       9. rem-exp-log 23.7%

          \[\leadsto \left|1 - \color{blue}{\left(1 + \sin th\right)}\right| \]

       10. +-commutative 23.7%

           \[\leadsto \left|1 - \color{blue}{\left(\sin th + 1\right)}\right| \]

   11. Applied egg-rr 23.7%

       \[\leadsto \color{blue}{\left|1 - \left(\sin th + 1\right)\right|} \]

   if -0.0040000000000000001 < (sin.f64 ky) < 1.0000000000000001e-110

   1. Initial program 86.6%

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

      1. unpow2 86.6%

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

      2. sqr-neg 86.6%

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

      3. sin-neg 86.6%

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

      4. sin-neg 86.6%

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

      5. unpow2 86.6%

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

      6. associate-*l/ 81.8%

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

      7. associate-/l* 86.6%

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

      8. +-commutative 86.6%

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

      9. unpow2 86.6%

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

      10. sin-neg 86.6%

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

      11. sin-neg 86.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 52.8%

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

      1. associate-/l* 54.5%

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

   7. Simplified 54.5%

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

   if 1.0000000000000001e-110 < (sin.f64 ky) < 9.9999999999999992e-72 or 1.00000000000000001e-50 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. +-commutative 99.7%

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

      9. unpow2 99.7%

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

      10. sin-neg 99.7%

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

      11. sin-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 60.0%

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

   if 9.9999999999999992e-72 < (sin.f64 ky) < 1.00000000000000001e-50

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0 44.2%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\left|1 + \left(-1 - \sin th\right)\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-110}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-71} \lor \neg \left(\sin ky \leq 10^{-50}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\left|1 + \left(-1 - \sin th\right)\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-110}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-71} \lor \neg \left(\sin ky \leq 10^{-50}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.004)
   (fabs (+ 1.0 (- -1.0 (sin th))))
   (if (<= (sin ky) 1e-110)
     (* ky (/ (sin th) (sin kx)))
     (if (or (<= (sin ky) 1e-71) (not (<= (sin ky) 1e-50)))
       (sin th)
       (/ (sin th) (/ (sin kx) ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.004) {
		tmp = fabs((1.0 + (-1.0 - sin(th))));
	} else if (sin(ky) <= 1e-110) {
		tmp = ky * (sin(th) / sin(kx));
	} else if ((sin(ky) <= 1e-71) || !(sin(ky) <= 1e-50)) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(kx) / ky);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.004d0)) then
        tmp = abs((1.0d0 + ((-1.0d0) - sin(th))))
    else if (sin(ky) <= 1d-110) then
        tmp = ky * (sin(th) / sin(kx))
    else if ((sin(ky) <= 1d-71) .or. (.not. (sin(ky) <= 1d-50))) then
        tmp = sin(th)
    else
        tmp = sin(th) / (sin(kx) / ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.004) {
		tmp = Math.abs((1.0 + (-1.0 - Math.sin(th))));
	} else if (Math.sin(ky) <= 1e-110) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else if ((Math.sin(ky) <= 1e-71) || !(Math.sin(ky) <= 1e-50)) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.004:
		tmp = math.fabs((1.0 + (-1.0 - math.sin(th))))
	elif math.sin(ky) <= 1e-110:
		tmp = ky * (math.sin(th) / math.sin(kx))
	elif (math.sin(ky) <= 1e-71) or not (math.sin(ky) <= 1e-50):
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.004)
		tmp = abs(Float64(1.0 + Float64(-1.0 - sin(th))));
	elseif (sin(ky) <= 1e-110)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	elseif ((sin(ky) <= 1e-71) || !(sin(ky) <= 1e-50))
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.004)
		tmp = abs((1.0 + (-1.0 - sin(th))));
	elseif (sin(ky) <= 1e-110)
		tmp = ky * (sin(th) / sin(kx));
	elseif ((sin(ky) <= 1e-71) || ~((sin(ky) <= 1e-50)))
		tmp = sin(th);
	else
		tmp = sin(th) / (sin(kx) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.004], N[Abs[N[(1.0 + N[(-1.0 - N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-110], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[Sin[ky], $MachinePrecision], 1e-71], N[Not[LessEqual[N[Sin[ky], $MachinePrecision], 1e-50]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.0040000000000000001

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. +-commutative 99.7%

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

      9. unpow2 99.7%

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

      10. sin-neg 99.7%

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

      11. sin-neg 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.6%

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

      1. expm1-log1p-u 2.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. expm1-undefine 3.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]

      3. associate-*r/ 3.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sin ky \cdot \sin th}{\sin ky}}\right)} - 1 \]

      4. *-commutative 3.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sin th \cdot \sin ky}}{\sin ky}\right)} - 1 \]

      5. associate-/l* 3.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sin th \cdot \frac{\sin ky}{\sin ky}}\right)} - 1 \]

   7. Applied egg-rr 3.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th \cdot \frac{\sin ky}{\sin ky}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-define 2.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th \cdot \frac{\sin ky}{\sin ky}\right)\right)} \]

      2. *-inverses 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin th \cdot \color{blue}{1}\right)\right) \]

      3. *-rgt-identity 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 2.6%

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

       2. add-sqr-sqrt 1.5%

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

       3. fabs-sqr 1.5%

          \[\leadsto \color{blue}{\left|\sqrt{\sin th} \cdot \sqrt{\sin th}\right|} \]

       4. add-sqr-sqrt 34.2%

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

       5. expm1-log1p-u 34.2%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)}\right| \]

       6. expm1-undefine 23.7%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1}\right| \]

       7. fabs-sub 23.7%

          \[\leadsto \color{blue}{\left|1 - e^{\mathsf{log1p}\left(\sin th\right)}\right|} \]

       8. log1p-undefine 23.7%

          \[\leadsto \left|1 - e^{\color{blue}{\log \left(1 + \sin th\right)}}\right| \]

       9. rem-exp-log 23.7%

          \[\leadsto \left|1 - \color{blue}{\left(1 + \sin th\right)}\right| \]

       10. +-commutative 23.7%

           \[\leadsto \left|1 - \color{blue}{\left(\sin th + 1\right)}\right| \]

   11. Applied egg-rr 23.7%

       \[\leadsto \color{blue}{\left|1 - \left(\sin th + 1\right)\right|} \]

   if -0.0040000000000000001 < (sin.f64 ky) < 1.0000000000000001e-110

   1. Initial program 86.6%

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

      1. unpow2 86.6%

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

      2. sqr-neg 86.6%

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

      3. sin-neg 86.6%

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

      4. sin-neg 86.6%

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

      5. unpow2 86.6%

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

      6. associate-*l/ 81.8%

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

      7. associate-/l* 86.6%

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

      8. +-commutative 86.6%

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

      9. unpow2 86.6%

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

      10. sin-neg 86.6%

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

      11. sin-neg 86.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 52.8%

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

      1. associate-/l* 54.5%

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

   7. Simplified 54.5%

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

   if 1.0000000000000001e-110 < (sin.f64 ky) < 9.9999999999999992e-72 or 1.00000000000000001e-50 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. +-commutative 99.7%

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

      9. unpow2 99.7%

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

      10. sin-neg 99.7%

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

      11. sin-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 60.0%

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

   if 9.9999999999999992e-72 < (sin.f64 ky) < 1.00000000000000001e-50

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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-undefine 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. clear-num 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in ky around 0 44.5%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\left|1 + \left(-1 - \sin th\right)\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-110}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-71} \lor \neg \left(\sin ky \leq 10^{-50}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\sin ky \cdot \left|\frac{th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-110}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-71} \lor \neg \left(\sin ky \leq 10^{-50}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.004)
   (* (sin ky) (fabs (/ th (sin ky))))
   (if (<= (sin ky) 1e-110)
     (* ky (/ (sin th) (sin kx)))
     (if (or (<= (sin ky) 1e-71) (not (<= (sin ky) 1e-50)))
       (sin th)
       (/ (sin th) (/ (sin kx) ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.004) {
		tmp = sin(ky) * fabs((th / sin(ky)));
	} else if (sin(ky) <= 1e-110) {
		tmp = ky * (sin(th) / sin(kx));
	} else if ((sin(ky) <= 1e-71) || !(sin(ky) <= 1e-50)) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(kx) / ky);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.004d0)) then
        tmp = sin(ky) * abs((th / sin(ky)))
    else if (sin(ky) <= 1d-110) then
        tmp = ky * (sin(th) / sin(kx))
    else if ((sin(ky) <= 1d-71) .or. (.not. (sin(ky) <= 1d-50))) then
        tmp = sin(th)
    else
        tmp = sin(th) / (sin(kx) / ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.004) {
		tmp = Math.sin(ky) * Math.abs((th / Math.sin(ky)));
	} else if (Math.sin(ky) <= 1e-110) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else if ((Math.sin(ky) <= 1e-71) || !(Math.sin(ky) <= 1e-50)) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.004:
		tmp = math.sin(ky) * math.fabs((th / math.sin(ky)))
	elif math.sin(ky) <= 1e-110:
		tmp = ky * (math.sin(th) / math.sin(kx))
	elif (math.sin(ky) <= 1e-71) or not (math.sin(ky) <= 1e-50):
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.004)
		tmp = Float64(sin(ky) * abs(Float64(th / sin(ky))));
	elseif (sin(ky) <= 1e-110)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	elseif ((sin(ky) <= 1e-71) || !(sin(ky) <= 1e-50))
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.004)
		tmp = sin(ky) * abs((th / sin(ky)));
	elseif (sin(ky) <= 1e-110)
		tmp = ky * (sin(th) / sin(kx));
	elseif ((sin(ky) <= 1e-71) || ~((sin(ky) <= 1e-50)))
		tmp = sin(th);
	else
		tmp = sin(th) / (sin(kx) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.004], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(th / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-110], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[Sin[ky], $MachinePrecision], 1e-71], N[Not[LessEqual[N[Sin[ky], $MachinePrecision], 1e-50]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.0040000000000000001

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. +-commutative 99.7%

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

      9. unpow2 99.7%

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

      10. sin-neg 99.7%

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

      11. sin-neg 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.6%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 23.2%

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

      3. pow2 23.2%

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

   7. Applied egg-rr 23.2%

      \[\leadsto \sin ky \cdot \color{blue}{\sqrt{{\left(\frac{\sin th}{\sin ky}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 23.2%

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

      2. rem-sqrt-square 35.9%

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

   9. Simplified 35.9%

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

   10. Taylor expanded in th around 0 22.7%

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

   if -0.0040000000000000001 < (sin.f64 ky) < 1.0000000000000001e-110

   1. Initial program 86.6%

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

      1. unpow2 86.6%

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

      2. sqr-neg 86.6%

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

      3. sin-neg 86.6%

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

      4. sin-neg 86.6%

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

      5. unpow2 86.6%

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

      6. associate-*l/ 81.8%

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

      7. associate-/l* 86.6%

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

      8. +-commutative 86.6%

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

      9. unpow2 86.6%

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

      10. sin-neg 86.6%

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

      11. sin-neg 86.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 52.8%

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

      1. associate-/l* 54.5%

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

   7. Simplified 54.5%

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

   if 1.0000000000000001e-110 < (sin.f64 ky) < 9.9999999999999992e-72 or 1.00000000000000001e-50 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. +-commutative 99.7%

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

      9. unpow2 99.7%

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

      10. sin-neg 99.7%

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

      11. sin-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 60.0%

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

   if 9.9999999999999992e-72 < (sin.f64 ky) < 1.00000000000000001e-50

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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-undefine 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. clear-num 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in ky around 0 44.5%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\sin ky \cdot \left|\frac{th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-110}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-71} \lor \neg \left(\sin ky \leq 10^{-50}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \mathbf{elif}\;\sin kx \leq 10^{-48}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.005)
   (/ (sin ky) (fabs (/ (sin kx) (sin th))))
   (if (<= (sin kx) 1e-48)
     (* (sin ky) (fabs (/ (sin th) (sin ky))))
     (* (sin ky) (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.005) {
		tmp = sin(ky) / fabs((sin(kx) / sin(th)));
	} else if (sin(kx) <= 1e-48) {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.005d0)) then
        tmp = sin(ky) / abs((sin(kx) / sin(th)))
    else if (sin(kx) <= 1d-48) then
        tmp = sin(ky) * abs((sin(th) / sin(ky)))
    else
        tmp = sin(ky) * (sin(th) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.005) {
		tmp = Math.sin(ky) / Math.abs((Math.sin(kx) / Math.sin(th)));
	} else if (Math.sin(kx) <= 1e-48) {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.005:
		tmp = math.sin(ky) / math.fabs((math.sin(kx) / math.sin(th)))
	elif math.sin(kx) <= 1e-48:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.005)
		tmp = Float64(sin(ky) / abs(Float64(sin(kx) / sin(th))));
	elseif (sin(kx) <= 1e-48)
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.005)
		tmp = sin(ky) / abs((sin(kx) / sin(th)));
	elseif (sin(kx) <= 1e-48)
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-48], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      6. associate-*l/ 99.3%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in ky around 0 19.9%

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

      1. add-sqr-sqrt 9.8%

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

      2. sqrt-unprod 40.8%

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

      3. pow2 40.8%

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

   9. Applied egg-rr 40.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\left(\frac{\sin kx}{\sin th}\right)}^{2}}}} \]
   10. Step-by-step derivation

       1. unpow2 40.8%

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

       2. rem-sqrt-square 41.5%

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

   11. Simplified 41.5%

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

   if -0.0050000000000000001 < (sin.f64 kx) < 9.9999999999999997e-49

   1. Initial program 88.5%

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

      1. unpow2 88.5%

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

      2. sqr-neg 88.5%

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

      3. sin-neg 88.5%

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

      4. sin-neg 88.5%

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

      5. unpow2 88.5%

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

      6. associate-*l/ 84.5%

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

      7. associate-/l* 88.4%

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

      8. +-commutative 88.4%

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

      9. unpow2 88.4%

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

      10. sin-neg 88.4%

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

      11. sin-neg 88.4%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 30.8%

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

      1. add-sqr-sqrt 12.7%

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

      2. sqrt-unprod 24.4%

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

      3. pow2 24.4%

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

   7. Applied egg-rr 24.4%

      \[\leadsto \sin ky \cdot \color{blue}{\sqrt{{\left(\frac{\sin th}{\sin ky}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 24.4%

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

      2. rem-sqrt-square 38.2%

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

   9. Simplified 38.2%

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

   if 9.9999999999999997e-49 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      6. associate-*l/ 99.2%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 65.3%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \mathbf{elif}\;\sin kx \leq 10^{-48}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.0033:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{ky \cdot \sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.0033)
   (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
   (/ 1.0 (/ (hypot (sin ky) (sin kx)) (* ky (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0033) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else {
		tmp = 1.0 / (hypot(sin(ky), sin(kx)) / (ky * sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0033) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else {
		tmp = 1.0 / (Math.hypot(Math.sin(ky), Math.sin(kx)) / (ky * Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.0033:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	else:
		tmp = 1.0 / (math.hypot(math.sin(ky), math.sin(kx)) / (ky * math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.0033)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	else
		tmp = Float64(1.0 / Float64(hypot(sin(ky), sin(kx)) / Float64(ky * sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.0033)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	else
		tmp = 1.0 / (hypot(sin(ky), sin(kx)) / (ky * sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.0033], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.0033

   1. Initial program 94.6%

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

      1. unpow2 94.6%

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

      2. sqr-neg 94.6%

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

      3. sin-neg 94.6%

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

      4. sin-neg 94.6%

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

      5. unpow2 94.6%

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

      6. associate-*l/ 91.9%

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

      7. associate-/l* 94.7%

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

      8. +-commutative 94.7%

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

      9. unpow2 94.7%

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

      10. sin-neg 94.7%

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

      11. sin-neg 94.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in th around 0 64.0%

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

      1. associate-*l/ 64.5%

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

      2. +-commutative 64.5%

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

      3. unpow2 64.5%

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

      4. unpow2 64.5%

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

      5. hypot-undefine 68.0%

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

      6. *-lft-identity 68.0%

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

      7. hypot-undefine 64.5%

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

      8. unpow2 64.5%

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

      9. unpow2 64.5%

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

      10. +-commutative 64.5%

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

      11. unpow2 64.5%

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

      12. unpow2 64.5%

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

      13. hypot-define 68.0%

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

   9. Simplified 68.0%

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

   if 0.0033 < th

   1. Initial program 91.9%

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

      1. unpow2 91.9%

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

      2. sqr-neg 91.9%

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

      3. sin-neg 91.9%

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

      4. sin-neg 91.9%

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

      5. unpow2 91.9%

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

      6. associate-*l/ 91.8%

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

      7. associate-/l* 91.9%

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

      8. +-commutative 91.9%

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

      9. unpow2 91.9%

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

      10. sin-neg 91.9%

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

      11. sin-neg 91.9%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in ky around 0 49.1%

      \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{ky \cdot \sin th}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.0033:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{ky \cdot \sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2.75 \cdot 10^{-129}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;ky \leq 4.8 \cdot 10^{+104} \lor \neg \left(ky \leq 3.5 \cdot 10^{+191}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin th\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 2.75e-129)
   (* ky (/ (sin th) kx))
   (if (or (<= ky 4.8e+104) (not (<= ky 3.5e+191))) (sin th) (fabs (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.75e-129) {
		tmp = ky * (sin(th) / kx);
	} else if ((ky <= 4.8e+104) || !(ky <= 3.5e+191)) {
		tmp = sin(th);
	} else {
		tmp = fabs(sin(th));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 2.75d-129) then
        tmp = ky * (sin(th) / kx)
    else if ((ky <= 4.8d+104) .or. (.not. (ky <= 3.5d+191))) then
        tmp = sin(th)
    else
        tmp = abs(sin(th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.75e-129) {
		tmp = ky * (Math.sin(th) / kx);
	} else if ((ky <= 4.8e+104) || !(ky <= 3.5e+191)) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.abs(Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 2.75e-129:
		tmp = ky * (math.sin(th) / kx)
	elif (ky <= 4.8e+104) or not (ky <= 3.5e+191):
		tmp = math.sin(th)
	else:
		tmp = math.fabs(math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 2.75e-129)
		tmp = Float64(ky * Float64(sin(th) / kx));
	elseif ((ky <= 4.8e+104) || !(ky <= 3.5e+191))
		tmp = sin(th);
	else
		tmp = abs(sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 2.75e-129)
		tmp = ky * (sin(th) / kx);
	elseif ((ky <= 4.8e+104) || ~((ky <= 3.5e+191)))
		tmp = sin(th);
	else
		tmp = abs(sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 2.75e-129], N[(ky * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[ky, 4.8e+104], N[Not[LessEqual[ky, 3.5e+191]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 2.75000000000000012e-129

   1. Initial program 91.4%

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

      1. unpow2 91.4%

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

      2. sqr-neg 91.4%

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

      3. sin-neg 91.4%

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

      4. sin-neg 91.4%

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

      5. unpow2 91.4%

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

      6. associate-*l/ 88.4%

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

      7. associate-/l* 91.4%

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

      8. +-commutative 91.4%

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

      9. unpow2 91.4%

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

      10. sin-neg 91.4%

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

      11. sin-neg 91.4%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 94.4%

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

      2. clear-num 94.1%

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

   6. Applied egg-rr 94.1%

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

   7. Taylor expanded in ky around 0 32.4%

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

   8. Taylor expanded in kx around 0 18.2%

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

      1. associate-/l* 19.2%

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

   10. Simplified 19.2%

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

   if 2.75000000000000012e-129 < ky < 4.8e104 or 3.4999999999999997e191 < ky

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.8%

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

      8. +-commutative 99.8%

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

      9. unpow2 99.8%

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

      10. sin-neg 99.8%

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

      11. sin-neg 99.8%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 28.6%

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

   if 4.8e104 < ky < 3.4999999999999997e191

   1. Initial program 99.9%

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

      1. unpow2 99.9%

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

      2. sqr-neg 99.9%

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

      3. sin-neg 99.9%

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

      4. sin-neg 99.9%

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

      5. unpow2 99.9%

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

      6. associate-*l/ 99.8%

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

      7. associate-/l* 99.9%

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

      8. +-commutative 99.9%

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

      9. unpow2 99.9%

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

      10. sin-neg 99.9%

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

      11. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in kx around 0 24.6%

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

      1. add-sqr-sqrt 7.2%

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

      2. sqrt-unprod 37.0%

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

      3. pow2 37.0%

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

      4. associate-*r/ 37.0%

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

      5. *-commutative 37.0%

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

      6. associate-/l* 37.1%

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

   7. Applied egg-rr 37.1%

      \[\leadsto \color{blue}{\sqrt{{\left(\sin th \cdot \frac{\sin ky}{\sin ky}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 37.1%

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

      2. rem-sqrt-square 43.3%

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

      3. *-inverses 43.3%

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

      4. *-rgt-identity 43.3%

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

   9. Simplified 43.3%

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

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 2.75 \cdot 10^{-129}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;ky \leq 4.8 \cdot 10^{+104} \lor \neg \left(ky \leq 3.5 \cdot 10^{+191}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin th\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 24.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 9.2 \cdot 10^{-111}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 9.2e-111) (* ky (/ th (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 9.2e-111) {
		tmp = ky * (th / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 9.2d-111) then
        tmp = ky * (th / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 9.2e-111) {
		tmp = ky * (th / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 9.2e-111:
		tmp = ky * (th / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 9.2e-111)
		tmp = Float64(ky * Float64(th / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 9.2e-111)
		tmp = ky * (th / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 9.2e-111], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 9.2e-111

   1. Initial program 91.6%

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

      1. unpow2 91.6%

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

      2. sqr-neg 91.6%

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

      3. sin-neg 91.6%

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

      4. sin-neg 91.6%

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

      5. unpow2 91.6%

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

      6. associate-*l/ 88.7%

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

      7. associate-/l* 91.6%

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

      8. +-commutative 91.6%

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

      9. unpow2 91.6%

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

      10. sin-neg 91.6%

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

      11. sin-neg 91.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 94.5%

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

      2. clear-num 94.2%

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

   6. Applied egg-rr 94.2%

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

   7. Taylor expanded in ky around 0 33.3%

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

   8. Taylor expanded in th around 0 19.9%

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

      1. associate-/l* 20.9%

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

   10. Simplified 20.9%

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

   if 9.2e-111 < ky

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 99.7%

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

      7. associate-/l* 99.8%

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

      8. +-commutative 99.8%

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

      9. unpow2 99.8%

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

      10. sin-neg 99.8%

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

      11. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 29.1%

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

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 9.2 \cdot 10^{-111}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 3.2 \cdot 10^{-129}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 3.2e-129) (* ky (/ (sin th) kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3.2e-129) {
		tmp = ky * (sin(th) / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 3.2d-129) then
        tmp = ky * (sin(th) / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3.2e-129) {
		tmp = ky * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 3.2e-129:
		tmp = ky * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 3.2e-129)
		tmp = Float64(ky * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 3.2e-129)
		tmp = ky * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 3.2e-129], N[(ky * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 3.2000000000000003e-129

   1. Initial program 91.4%

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

      1. unpow2 91.4%

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

      2. sqr-neg 91.4%

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

      3. sin-neg 91.4%

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

      4. sin-neg 91.4%

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

      5. unpow2 91.4%

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

      6. associate-*l/ 88.4%

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

      7. associate-/l* 91.4%

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

      8. +-commutative 91.4%

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

      9. unpow2 91.4%

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

      10. sin-neg 91.4%

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

      11. sin-neg 91.4%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 94.4%

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

      2. clear-num 94.1%

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

   6. Applied egg-rr 94.1%

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

   7. Taylor expanded in ky around 0 32.4%

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

   8. Taylor expanded in kx around 0 18.2%

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

      1. associate-/l* 19.2%

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

   10. Simplified 19.2%

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

   if 3.2000000000000003e-129 < ky

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.7%

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

      7. associate-/l* 99.8%

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

      8. +-commutative 99.8%

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

      9. unpow2 99.8%

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

      10. sin-neg 99.8%

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

      11. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 27.9%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 3.2 \cdot 10^{-129}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 24.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 4e-7) (sin th) (+ (+ th 1.0) -1.0)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4e-7) {
		tmp = sin(th);
	} else {
		tmp = (th + 1.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 4d-7) then
        tmp = sin(th)
    else
        tmp = (th + 1.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4e-7) {
		tmp = Math.sin(th);
	} else {
		tmp = (th + 1.0) + -1.0;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 4e-7:
		tmp = math.sin(th)
	else:
		tmp = (th + 1.0) + -1.0
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 4e-7)
		tmp = sin(th);
	else
		tmp = Float64(Float64(th + 1.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 4e-7)
		tmp = sin(th);
	else
		tmp = (th + 1.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 4e-7], N[Sin[th], $MachinePrecision], N[(N[(th + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 3.9999999999999998e-7

   1. Initial program 92.4%

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

      1. unpow2 92.4%

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

      2. sqr-neg 92.4%

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

      3. sin-neg 92.4%

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

      4. sin-neg 92.4%

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

      5. unpow2 92.4%

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

      6. associate-*l/ 89.7%

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

      7. associate-/l* 92.3%

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

      8. +-commutative 92.3%

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

      9. unpow2 92.3%

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

      10. sin-neg 92.3%

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

      11. sin-neg 92.3%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 24.0%

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

   if 3.9999999999999998e-7 < kx

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. sqr-neg 99.3%

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

      3. sin-neg 99.3%

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

      4. sin-neg 99.3%

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

      5. unpow2 99.3%

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

      6. associate-*l/ 99.1%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 8.0%

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

      1. expm1-log1p-u 8.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. expm1-undefine 18.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]

      3. associate-*r/ 18.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sin ky \cdot \sin th}{\sin ky}}\right)} - 1 \]

      4. *-commutative 18.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sin th \cdot \sin ky}}{\sin ky}\right)} - 1 \]

      5. associate-/l* 18.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sin th \cdot \frac{\sin ky}{\sin ky}}\right)} - 1 \]

   7. Applied egg-rr 18.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th \cdot \frac{\sin ky}{\sin ky}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-define 8.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th \cdot \frac{\sin ky}{\sin ky}\right)\right)} \]

      2. *-inverses 8.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin th \cdot \color{blue}{1}\right)\right) \]

      3. *-rgt-identity 8.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 8.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 18.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 18.3%

          \[\leadsto e^{\color{blue}{\log \left(1 + \sin th\right)}} - 1 \]

       3. rem-exp-log 18.3%

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

       4. +-commutative 18.3%

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

   11. Applied egg-rr 18.3%

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

   12. Taylor expanded in th around 0 16.5%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. +-commutative 16.5%

          \[\leadsto \color{blue}{\left(th + 1\right)} - 1 \]

   14. Simplified 16.5%

       \[\leadsto \color{blue}{\left(th + 1\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 15.8% accurate, 70.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 8.8 \cdot 10^{-10}:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 8.8e-10) th (+ (+ th 1.0) -1.0)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8.8e-10) {
		tmp = th;
	} else {
		tmp = (th + 1.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 8.8d-10) then
        tmp = th
    else
        tmp = (th + 1.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8.8e-10) {
		tmp = th;
	} else {
		tmp = (th + 1.0) + -1.0;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 8.8e-10:
		tmp = th
	else:
		tmp = (th + 1.0) + -1.0
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 8.8e-10)
		tmp = th;
	else
		tmp = Float64(Float64(th + 1.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 8.8e-10)
		tmp = th;
	else
		tmp = (th + 1.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 8.8e-10], th, N[(N[(th + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 8.7999999999999996e-10

   1. Initial program 92.4%

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

      1. unpow2 92.4%

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

      2. sqr-neg 92.4%

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

      3. sin-neg 92.4%

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

      4. sin-neg 92.4%

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

      5. unpow2 92.4%

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

      6. associate-*l/ 89.7%

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

      7. associate-/l* 92.3%

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

      8. +-commutative 92.3%

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

      9. unpow2 92.3%

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

      10. sin-neg 92.3%

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

      11. sin-neg 92.3%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 23.9%

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

   6. Taylor expanded in th around 0 15.7%

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

   if 8.7999999999999996e-10 < kx

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. sqr-neg 99.3%

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

      3. sin-neg 99.3%

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

      4. sin-neg 99.3%

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

      5. unpow2 99.3%

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

      6. associate-*l/ 99.1%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 8.0%

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

      1. expm1-log1p-u 8.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. expm1-undefine 18.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]

      3. associate-*r/ 18.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sin ky \cdot \sin th}{\sin ky}}\right)} - 1 \]

      4. *-commutative 18.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sin th \cdot \sin ky}}{\sin ky}\right)} - 1 \]

      5. associate-/l* 18.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sin th \cdot \frac{\sin ky}{\sin ky}}\right)} - 1 \]

   7. Applied egg-rr 18.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th \cdot \frac{\sin ky}{\sin ky}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-define 8.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th \cdot \frac{\sin ky}{\sin ky}\right)\right)} \]

      2. *-inverses 8.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin th \cdot \color{blue}{1}\right)\right) \]

      3. *-rgt-identity 8.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 8.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 18.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 18.3%

          \[\leadsto e^{\color{blue}{\log \left(1 + \sin th\right)}} - 1 \]

       3. rem-exp-log 18.3%

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

       4. +-commutative 18.3%

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

   11. Applied egg-rr 18.3%

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

   12. Taylor expanded in th around 0 16.5%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. +-commutative 16.5%

          \[\leadsto \color{blue}{\left(th + 1\right)} - 1 \]

   14. Simplified 16.5%

       \[\leadsto \color{blue}{\left(th + 1\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 8.8 \cdot 10^{-10}:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 13.2% 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 94.0%

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

   1. unpow2 94.0%

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

   2. sqr-neg 94.0%

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

   3. sin-neg 94.0%

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

   4. sin-neg 94.0%

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

   5. unpow2 94.0%

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

   6. associate-*l/ 91.9%

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

   7. associate-/l* 94.0%

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

   8. +-commutative 94.0%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin \left(-kx\right)}^{2}}}} \]

   9. unpow2 94.0%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)}}} \]

   10. sin-neg 94.0%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right)}} \]

   11. sin-neg 94.0%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)}}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
4. Add Preprocessing

5. Taylor expanded in kx around 0 20.2%

   \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]

6. Taylor expanded in th around 0 13.5%

   \[\leadsto \color{blue}{th} \]

7. Final simplification 13.5%

   \[\leadsto th \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(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)))