Toniolo and Linder, Equation (3b), real

Percentage Accurate: 60.9% → 75.8%

Time: 33.2s

Alternatives: 20

Speedup: 1.2×

• could not determine a ground truth

  1. kx = 1.3709716313874032e-63
  2. ky = 1.8904377355315626e-23
  3. th = 1.0356248581863755e+139

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: 60.9% 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: 75.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;ky \leq -2.6:\\ \;\;\;\;\left|\frac{\sin th}{\frac{t_1}{\sin ky}}\right|\\ \mathbf{elif}\;ky \leq 75000000:\\ \;\;\;\;\frac{\sin ky \cdot \left|\sin th\right|}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\left|\sin ky\right|}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= ky -2.6)
     (fabs (/ (sin th) (/ t_1 (sin ky))))
     (if (<= ky 75000000.0)
       (/ (* (sin ky) (fabs (sin th))) t_1)
       (* (sin th) (/ (fabs (sin ky)) t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (ky <= -2.6) {
		tmp = fabs((sin(th) / (t_1 / sin(ky))));
	} else if (ky <= 75000000.0) {
		tmp = (sin(ky) * fabs(sin(th))) / t_1;
	} else {
		tmp = sin(th) * (fabs(sin(ky)) / t_1);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (ky <= -2.6) {
		tmp = Math.abs((Math.sin(th) / (t_1 / Math.sin(ky))));
	} else if (ky <= 75000000.0) {
		tmp = (Math.sin(ky) * Math.abs(Math.sin(th))) / t_1;
	} else {
		tmp = Math.sin(th) * (Math.abs(Math.sin(ky)) / t_1);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if ky <= -2.6:
		tmp = math.fabs((math.sin(th) / (t_1 / math.sin(ky))))
	elif ky <= 75000000.0:
		tmp = (math.sin(ky) * math.fabs(math.sin(th))) / t_1
	else:
		tmp = math.sin(th) * (math.fabs(math.sin(ky)) / t_1)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (ky <= -2.6)
		tmp = abs(Float64(sin(th) / Float64(t_1 / sin(ky))));
	elseif (ky <= 75000000.0)
		tmp = Float64(Float64(sin(ky) * abs(sin(th))) / t_1);
	else
		tmp = Float64(sin(th) * Float64(abs(sin(ky)) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (ky <= -2.6)
		tmp = abs((sin(th) / (t_1 / sin(ky))));
	elseif (ky <= 75000000.0)
		tmp = (sin(ky) * abs(sin(th))) / t_1;
	else
		tmp = sin(th) * (abs(sin(ky)) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[ky, -2.6], N[Abs[N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ky, 75000000.0], N[(N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -2.60000000000000009

   1. Initial program 50.2%

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

      1. +-commutative 50.2%

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

      2. unpow2 50.2%

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

      3. unpow2 50.2%

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

      4. hypot-def 50.2%

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

   3. Simplified 50.2%

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

      1. add-sqr-sqrt 42.8%

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

      2. sqrt-unprod 63.0%

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

      3. pow2 63.0%

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

      4. associate-/r/ 63.0%

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

      5. div-inv 62.9%

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

      6. clear-num 63.1%

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

   5. Applied egg-rr 63.1%

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

      1. unpow2 63.1%

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

      2. rem-sqrt-square 80.0%

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

      3. *-commutative 80.0%

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

      4. associate-/r/ 80.0%

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

   7. Simplified 80.0%

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

   if -2.60000000000000009 < ky < 7.5e7

   1. Initial program 71.3%

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

      1. associate-*l/ 68.9%

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

      2. +-commutative 68.9%

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

      3. unpow2 68.9%

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

      4. unpow2 68.9%

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

      5. hypot-def 70.8%

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

   3. Simplified 70.8%

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

      1. add-sqr-sqrt 50.9%

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

      2. sqrt-unprod 82.4%

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

      3. pow2 82.4%

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

   5. Applied egg-rr 82.4%

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

      1. unpow2 82.4%

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

      2. rem-sqrt-square 83.1%

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

   7. Simplified 83.1%

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

   if 7.5e7 < ky

   1. Initial program 44.1%

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

      1. +-commutative 44.1%

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

      2. unpow2 44.1%

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

      3. unpow2 44.1%

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

      4. hypot-def 44.1%

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

   3. Simplified 44.1%

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

      1. add-sqr-sqrt 32.4%

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

      2. sqrt-prod 82.6%

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

      3. rem-sqrt-square 82.6%

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

   5. Applied egg-rr 82.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -2.6:\\ \;\;\;\;\left|\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\right|\\ \mathbf{elif}\;ky \leq 75000000:\\ \;\;\;\;\frac{\sin ky \cdot \left|\sin th\right|}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\left|\sin ky\right|}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]

Alternative 2: 67.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \left|\sin ky\right|\\ \mathbf{if}\;\sin th \leq -0.86:\\ \;\;\;\;\sin th \cdot \frac{t_2}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq 3.2 \cdot 10^{-139}:\\ \;\;\;\;th \cdot \frac{t_2}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))) (t_2 (fabs (sin ky))))
   (if (<= (sin th) -0.86)
     (* (sin th) (/ t_2 (hypot ky (sin kx))))
     (if (<= (sin th) -0.02)
       (fabs (sin th))
       (if (<= (sin th) 3.2e-139)
         (* th (/ t_2 t_1))
         (* (sin ky) (/ (sin th) t_1)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = fabs(sin(ky));
	double tmp;
	if (sin(th) <= -0.86) {
		tmp = sin(th) * (t_2 / hypot(ky, sin(kx)));
	} else if (sin(th) <= -0.02) {
		tmp = fabs(sin(th));
	} else if (sin(th) <= 3.2e-139) {
		tmp = th * (t_2 / t_1);
	} else {
		tmp = sin(ky) * (sin(th) / t_1);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double t_2 = Math.abs(Math.sin(ky));
	double tmp;
	if (Math.sin(th) <= -0.86) {
		tmp = Math.sin(th) * (t_2 / Math.hypot(ky, Math.sin(kx)));
	} else if (Math.sin(th) <= -0.02) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(th) <= 3.2e-139) {
		tmp = th * (t_2 / t_1);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / t_1);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	t_2 = math.fabs(math.sin(ky))
	tmp = 0
	if math.sin(th) <= -0.86:
		tmp = math.sin(th) * (t_2 / math.hypot(ky, math.sin(kx)))
	elif math.sin(th) <= -0.02:
		tmp = math.fabs(math.sin(th))
	elif math.sin(th) <= 3.2e-139:
		tmp = th * (t_2 / t_1)
	else:
		tmp = math.sin(ky) * (math.sin(th) / t_1)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = abs(sin(ky))
	tmp = 0.0
	if (sin(th) <= -0.86)
		tmp = Float64(sin(th) * Float64(t_2 / hypot(ky, sin(kx))));
	elseif (sin(th) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(th) <= 3.2e-139)
		tmp = Float64(th * Float64(t_2 / t_1));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	t_2 = abs(sin(ky));
	tmp = 0.0;
	if (sin(th) <= -0.86)
		tmp = sin(th) * (t_2 / hypot(ky, sin(kx)));
	elseif (sin(th) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(th) <= 3.2e-139)
		tmp = th * (t_2 / t_1);
	else
		tmp = sin(ky) * (sin(th) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Sin[th], $MachinePrecision], -0.86], N[(N[Sin[th], $MachinePrecision] * N[(t$95$2 / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], -0.02], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 3.2e-139], N[(th * N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 th) < -0.859999999999999987

   1. Initial program 26.7%

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

      1. +-commutative 26.7%

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

      2. unpow2 26.7%

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

      3. unpow2 26.7%

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

      4. hypot-def 26.7%

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

   3. Simplified 26.7%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-prod 23.7%

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

      3. rem-sqrt-square 47.0%

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

   5. Applied egg-rr 47.0%

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

   6. Taylor expanded in ky around 0 44.0%

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

   if -0.859999999999999987 < (sin.f64 th) < -0.0200000000000000004

   1. Initial program 39.0%

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

      1. associate-*l/ 39.0%

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

      2. +-commutative 39.0%

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

      3. unpow2 39.0%

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

      4. unpow2 39.0%

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

      5. hypot-def 39.1%

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

   3. Simplified 39.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 62.4%

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

      3. pow2 62.4%

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

   5. Applied egg-rr 62.4%

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

      1. unpow2 62.4%

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

      2. rem-sqrt-square 62.4%

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

   7. Simplified 62.4%

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

   8. Taylor expanded in kx around 0 42.2%

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

   if -0.0200000000000000004 < (sin.f64 th) < 3.1999999999999999e-139

   1. Initial program 62.4%

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

      1. +-commutative 62.4%

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

      2. unpow2 62.4%

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

      3. unpow2 62.4%

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

      4. hypot-def 64.4%

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

   3. Simplified 64.4%

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

      1. add-sqr-sqrt 43.7%

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

      2. sqrt-prod 88.0%

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

      3. rem-sqrt-square 89.8%

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

   5. Applied egg-rr 89.8%

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

   6. Taylor expanded in th around 0 89.5%

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

   if 3.1999999999999999e-139 < (sin.f64 th)

   1. Initial program 69.4%

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

      1. associate-*l/ 69.4%

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

      2. +-commutative 69.4%

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

      3. unpow2 69.4%

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

      4. unpow2 69.4%

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

      5. hypot-def 71.5%

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

   3. Simplified 71.5%

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

      1. associate-/l* 71.5%

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

      2. div-inv 71.4%

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

      3. clear-num 71.6%

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

   5. Applied egg-rr 71.6%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.86:\\ \;\;\;\;\sin th \cdot \frac{\left|\sin ky\right|}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq 3.2 \cdot 10^{-139}:\\ \;\;\;\;th \cdot \frac{\left|\sin ky\right|}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]

Alternative 3: 75.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \left|\frac{\sin th}{\frac{t_1}{\sin ky}}\right|\\ \mathbf{if}\;th \leq -102000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 3.2 \cdot 10^{-139}:\\ \;\;\;\;th \cdot \frac{\left|\sin ky\right|}{t_1}\\ \mathbf{elif}\;th \leq 1.32 \cdot 10^{+66}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (fabs (/ (sin th) (/ t_1 (sin ky))))))
   (if (<= th -102000000.0)
     t_2
     (if (<= th 3.2e-139)
       (* th (/ (fabs (sin ky)) t_1))
       (if (<= th 1.32e+66) (* (sin ky) (/ (sin th) t_1)) t_2)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = fabs((sin(th) / (t_1 / sin(ky))));
	double tmp;
	if (th <= -102000000.0) {
		tmp = t_2;
	} else if (th <= 3.2e-139) {
		tmp = th * (fabs(sin(ky)) / t_1);
	} else if (th <= 1.32e+66) {
		tmp = sin(ky) * (sin(th) / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double t_2 = Math.abs((Math.sin(th) / (t_1 / Math.sin(ky))));
	double tmp;
	if (th <= -102000000.0) {
		tmp = t_2;
	} else if (th <= 3.2e-139) {
		tmp = th * (Math.abs(Math.sin(ky)) / t_1);
	} else if (th <= 1.32e+66) {
		tmp = Math.sin(ky) * (Math.sin(th) / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	t_2 = math.fabs((math.sin(th) / (t_1 / math.sin(ky))))
	tmp = 0
	if th <= -102000000.0:
		tmp = t_2
	elif th <= 3.2e-139:
		tmp = th * (math.fabs(math.sin(ky)) / t_1)
	elif th <= 1.32e+66:
		tmp = math.sin(ky) * (math.sin(th) / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = abs(Float64(sin(th) / Float64(t_1 / sin(ky))))
	tmp = 0.0
	if (th <= -102000000.0)
		tmp = t_2;
	elseif (th <= 3.2e-139)
		tmp = Float64(th * Float64(abs(sin(ky)) / t_1));
	elseif (th <= 1.32e+66)
		tmp = Float64(sin(ky) * Float64(sin(th) / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	t_2 = abs((sin(th) / (t_1 / sin(ky))));
	tmp = 0.0;
	if (th <= -102000000.0)
		tmp = t_2;
	elseif (th <= 3.2e-139)
		tmp = th * (abs(sin(ky)) / t_1);
	elseif (th <= 1.32e+66)
		tmp = sin(ky) * (sin(th) / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[th, -102000000.0], t$95$2, If[LessEqual[th, 3.2e-139], N[(th * N[(N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 1.32e+66], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if th < -1.02e8 or 1.32000000000000009e66 < th

   1. Initial program 47.5%

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

      1. +-commutative 47.5%

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

      2. unpow2 47.5%

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

      3. unpow2 47.5%

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

      4. hypot-def 48.4%

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

   3. Simplified 48.4%

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

      1. add-sqr-sqrt 33.2%

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

      2. sqrt-unprod 63.2%

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

      3. pow2 63.2%

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

      4. associate-/r/ 63.1%

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

      5. div-inv 63.1%

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

      6. clear-num 63.2%

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

   5. Applied egg-rr 63.2%

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

      1. unpow2 63.2%

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

      2. rem-sqrt-square 67.8%

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

      3. *-commutative 67.8%

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

      4. associate-/r/ 67.8%

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

   7. Simplified 67.8%

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

   if -1.02e8 < th < 3.1999999999999999e-139

   1. Initial program 62.2%

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

      1. +-commutative 62.2%

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

      2. unpow2 62.2%

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

      3. unpow2 62.2%

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

      4. hypot-def 64.1%

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

   3. Simplified 64.1%

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

      1. add-sqr-sqrt 43.8%

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

      2. sqrt-prod 88.2%

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

      3. rem-sqrt-square 89.9%

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

   5. Applied egg-rr 89.9%

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

   6. Taylor expanded in th around 0 87.9%

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

   if 3.1999999999999999e-139 < th < 1.32000000000000009e66

   1. Initial program 77.8%

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

      1. associate-*l/ 77.8%

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

      2. +-commutative 77.8%

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

      3. unpow2 77.8%

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

      4. unpow2 77.8%

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

      5. hypot-def 80.5%

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

   3. Simplified 80.5%

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

      1. associate-/l* 80.5%

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

      2. div-inv 80.4%

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

      3. clear-num 80.7%

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

   5. Applied egg-rr 80.7%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq -102000000:\\ \;\;\;\;\left|\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\right|\\ \mathbf{elif}\;th \leq 3.2 \cdot 10^{-139}:\\ \;\;\;\;th \cdot \frac{\left|\sin ky\right|}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 1.32 \cdot 10^{+66}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\right|\\ \end{array} \]

Alternative 4: 73.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin th}{t_1}\\ \mathbf{if}\;ky \leq -3.2 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{\sin th}{\frac{t_1}{\sin ky}}\right|\\ \mathbf{elif}\;ky \leq -1.6 \cdot 10^{-233}:\\ \;\;\;\;\sin ky \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;\left|\sin ky\right| \cdot t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))) (t_2 (/ (sin th) t_1)))
   (if (<= ky -3.2e+15)
     (fabs (/ (sin th) (/ t_1 (sin ky))))
     (if (<= ky -1.6e-233) (* (sin ky) t_2) (* (fabs (sin ky)) t_2)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = sin(th) / t_1;
	double tmp;
	if (ky <= -3.2e+15) {
		tmp = fabs((sin(th) / (t_1 / sin(ky))));
	} else if (ky <= -1.6e-233) {
		tmp = sin(ky) * t_2;
	} else {
		tmp = fabs(sin(ky)) * t_2;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double t_2 = Math.sin(th) / t_1;
	double tmp;
	if (ky <= -3.2e+15) {
		tmp = Math.abs((Math.sin(th) / (t_1 / Math.sin(ky))));
	} else if (ky <= -1.6e-233) {
		tmp = Math.sin(ky) * t_2;
	} else {
		tmp = Math.abs(Math.sin(ky)) * t_2;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	t_2 = math.sin(th) / t_1
	tmp = 0
	if ky <= -3.2e+15:
		tmp = math.fabs((math.sin(th) / (t_1 / math.sin(ky))))
	elif ky <= -1.6e-233:
		tmp = math.sin(ky) * t_2
	else:
		tmp = math.fabs(math.sin(ky)) * t_2
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(sin(th) / t_1)
	tmp = 0.0
	if (ky <= -3.2e+15)
		tmp = abs(Float64(sin(th) / Float64(t_1 / sin(ky))));
	elseif (ky <= -1.6e-233)
		tmp = Float64(sin(ky) * t_2);
	else
		tmp = Float64(abs(sin(ky)) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	t_2 = sin(th) / t_1;
	tmp = 0.0;
	if (ky <= -3.2e+15)
		tmp = abs((sin(th) / (t_1 / sin(ky))));
	elseif (ky <= -1.6e-233)
		tmp = sin(ky) * t_2;
	else
		tmp = abs(sin(ky)) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[ky, -3.2e+15], N[Abs[N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ky, -1.6e-233], N[(N[Sin[ky], $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -3.2e15

   1. Initial program 47.7%

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

      1. +-commutative 47.7%

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

      2. unpow2 47.7%

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

      3. unpow2 47.7%

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

      4. hypot-def 47.7%

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

   3. Simplified 47.7%

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

      1. add-sqr-sqrt 40.0%

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

      2. sqrt-unprod 64.3%

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

      3. pow2 64.3%

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

      4. associate-/r/ 64.2%

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

      5. div-inv 64.2%

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

      6. clear-num 64.4%

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

   5. Applied egg-rr 64.4%

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

      1. unpow2 64.4%

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

      2. rem-sqrt-square 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-/r/ 78.9%

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

   7. Simplified 78.9%

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

   if -3.2e15 < ky < -1.5999999999999999e-233

   1. Initial program 72.9%

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

      1. associate-*l/ 71.2%

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

      2. +-commutative 71.2%

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

      3. unpow2 71.2%

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

      4. unpow2 71.2%

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

      5. hypot-def 74.5%

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

   3. Simplified 74.5%

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

      1. associate-/l* 76.2%

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

      2. div-inv 75.9%

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

      3. clear-num 76.2%

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

   5. Applied egg-rr 76.2%

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

   if -1.5999999999999999e-233 < ky

   1. Initial program 55.5%

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

      1. associate-*l/ 54.2%

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

      2. +-commutative 54.2%

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

      3. unpow2 54.2%

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

      4. unpow2 54.2%

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

      5. hypot-def 54.4%

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

   3. Simplified 54.4%

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

      1. associate-/l* 56.9%

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

      2. div-inv 56.9%

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

      3. clear-num 57.0%

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

   5. Applied egg-rr 57.0%

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

      1. add-sqr-sqrt 41.9%

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

      2. sqrt-prod 70.0%

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

      3. rem-sqrt-square 80.1%

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

   7. Applied egg-rr 80.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -3.2 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\right|\\ \mathbf{elif}\;ky \leq -1.6 \cdot 10^{-233}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\sin ky\right| \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]

Alternative 5: 73.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;ky \leq -3.2 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{\sin th}{\frac{t_1}{\sin ky}}\right|\\ \mathbf{elif}\;ky \leq -1.6 \cdot 10^{-233}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\left|\sin ky\right|}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= ky -3.2e+15)
     (fabs (/ (sin th) (/ t_1 (sin ky))))
     (if (<= ky -1.6e-233)
       (* (sin ky) (/ (sin th) t_1))
       (* (sin th) (/ (fabs (sin ky)) t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (ky <= -3.2e+15) {
		tmp = fabs((sin(th) / (t_1 / sin(ky))));
	} else if (ky <= -1.6e-233) {
		tmp = sin(ky) * (sin(th) / t_1);
	} else {
		tmp = sin(th) * (fabs(sin(ky)) / t_1);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (ky <= -3.2e+15) {
		tmp = Math.abs((Math.sin(th) / (t_1 / Math.sin(ky))));
	} else if (ky <= -1.6e-233) {
		tmp = Math.sin(ky) * (Math.sin(th) / t_1);
	} else {
		tmp = Math.sin(th) * (Math.abs(Math.sin(ky)) / t_1);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if ky <= -3.2e+15:
		tmp = math.fabs((math.sin(th) / (t_1 / math.sin(ky))))
	elif ky <= -1.6e-233:
		tmp = math.sin(ky) * (math.sin(th) / t_1)
	else:
		tmp = math.sin(th) * (math.fabs(math.sin(ky)) / t_1)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (ky <= -3.2e+15)
		tmp = abs(Float64(sin(th) / Float64(t_1 / sin(ky))));
	elseif (ky <= -1.6e-233)
		tmp = Float64(sin(ky) * Float64(sin(th) / t_1));
	else
		tmp = Float64(sin(th) * Float64(abs(sin(ky)) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (ky <= -3.2e+15)
		tmp = abs((sin(th) / (t_1 / sin(ky))));
	elseif (ky <= -1.6e-233)
		tmp = sin(ky) * (sin(th) / t_1);
	else
		tmp = sin(th) * (abs(sin(ky)) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[ky, -3.2e+15], N[Abs[N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ky, -1.6e-233], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -3.2e15

   1. Initial program 47.7%

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

      1. +-commutative 47.7%

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

      2. unpow2 47.7%

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

      3. unpow2 47.7%

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

      4. hypot-def 47.7%

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

   3. Simplified 47.7%

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

      1. add-sqr-sqrt 40.0%

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

      2. sqrt-unprod 64.3%

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

      3. pow2 64.3%

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

      4. associate-/r/ 64.2%

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

      5. div-inv 64.2%

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

      6. clear-num 64.4%

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

   5. Applied egg-rr 64.4%

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

      1. unpow2 64.4%

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

      2. rem-sqrt-square 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-/r/ 78.9%

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

   7. Simplified 78.9%

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

   if -3.2e15 < ky < -1.5999999999999999e-233

   1. Initial program 72.9%

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

      1. associate-*l/ 71.2%

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

      2. +-commutative 71.2%

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

      3. unpow2 71.2%

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

      4. unpow2 71.2%

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

      5. hypot-def 74.5%

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

   3. Simplified 74.5%

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

      1. associate-/l* 76.2%

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

      2. div-inv 75.9%

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

      3. clear-num 76.2%

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

   5. Applied egg-rr 76.2%

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

   if -1.5999999999999999e-233 < ky

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      2. unpow2 55.5%

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

      3. unpow2 55.5%

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

      4. hypot-def 57.0%

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

   3. Simplified 57.0%

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

      1. add-sqr-sqrt 41.9%

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

      2. sqrt-prod 70.0%

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

      3. rem-sqrt-square 80.1%

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

   5. Applied egg-rr 80.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -3.2 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\right|\\ \mathbf{elif}\;ky \leq -1.6 \cdot 10^{-233}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\left|\sin ky\right|}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]

Alternative 6: 77.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;ky \leq -2.6:\\ \;\;\;\;\left|\frac{\sin th}{\frac{t_1}{\sin ky}}\right|\\ \mathbf{elif}\;ky \leq 170000000:\\ \;\;\;\;\left|\sin th\right| \cdot \frac{\sin ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\left|\sin ky\right|}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= ky -2.6)
     (fabs (/ (sin th) (/ t_1 (sin ky))))
     (if (<= ky 170000000.0)
       (* (fabs (sin th)) (/ (sin ky) t_1))
       (* (sin th) (/ (fabs (sin ky)) t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (ky <= -2.6) {
		tmp = fabs((sin(th) / (t_1 / sin(ky))));
	} else if (ky <= 170000000.0) {
		tmp = fabs(sin(th)) * (sin(ky) / t_1);
	} else {
		tmp = sin(th) * (fabs(sin(ky)) / t_1);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (ky <= -2.6) {
		tmp = Math.abs((Math.sin(th) / (t_1 / Math.sin(ky))));
	} else if (ky <= 170000000.0) {
		tmp = Math.abs(Math.sin(th)) * (Math.sin(ky) / t_1);
	} else {
		tmp = Math.sin(th) * (Math.abs(Math.sin(ky)) / t_1);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if ky <= -2.6:
		tmp = math.fabs((math.sin(th) / (t_1 / math.sin(ky))))
	elif ky <= 170000000.0:
		tmp = math.fabs(math.sin(th)) * (math.sin(ky) / t_1)
	else:
		tmp = math.sin(th) * (math.fabs(math.sin(ky)) / t_1)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (ky <= -2.6)
		tmp = abs(Float64(sin(th) / Float64(t_1 / sin(ky))));
	elseif (ky <= 170000000.0)
		tmp = Float64(abs(sin(th)) * Float64(sin(ky) / t_1));
	else
		tmp = Float64(sin(th) * Float64(abs(sin(ky)) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (ky <= -2.6)
		tmp = abs((sin(th) / (t_1 / sin(ky))));
	elseif (ky <= 170000000.0)
		tmp = abs(sin(th)) * (sin(ky) / t_1);
	else
		tmp = sin(th) * (abs(sin(ky)) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[ky, -2.6], N[Abs[N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ky, 170000000.0], N[(N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -2.60000000000000009

   1. Initial program 50.2%

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

      1. +-commutative 50.2%

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

      2. unpow2 50.2%

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

      3. unpow2 50.2%

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

      4. hypot-def 50.2%

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

   3. Simplified 50.2%

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

      1. add-sqr-sqrt 42.8%

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

      2. sqrt-unprod 63.0%

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

      3. pow2 63.0%

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

      4. associate-/r/ 63.0%

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

      5. div-inv 62.9%

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

      6. clear-num 63.1%

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

   5. Applied egg-rr 63.1%

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

      1. unpow2 63.1%

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

      2. rem-sqrt-square 80.0%

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

      3. *-commutative 80.0%

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

      4. associate-/r/ 80.0%

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

   7. Simplified 80.0%

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

   if -2.60000000000000009 < ky < 1.7e8

   1. Initial program 71.3%

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

      1. +-commutative 71.3%

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

      2. unpow2 71.3%

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

      3. unpow2 71.3%

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

      4. hypot-def 74.8%

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

   3. Simplified 74.8%

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

      1. add-sqr-sqrt 50.9%

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

      2. sqrt-unprod 82.4%

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

      3. pow2 82.4%

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

   5. Applied egg-rr 82.5%

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

      1. unpow2 82.4%

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

      2. rem-sqrt-square 83.1%

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

   7. Simplified 83.1%

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

   if 1.7e8 < ky

   1. Initial program 44.1%

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

      1. +-commutative 44.1%

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

      2. unpow2 44.1%

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

      3. unpow2 44.1%

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

      4. hypot-def 44.1%

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

   3. Simplified 44.1%

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

      1. add-sqr-sqrt 32.4%

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

      2. sqrt-prod 82.6%

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

      3. rem-sqrt-square 82.6%

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

   5. Applied egg-rr 82.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -2.6:\\ \;\;\;\;\left|\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\right|\\ \mathbf{elif}\;ky \leq 170000000:\\ \;\;\;\;\left|\sin th\right| \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\left|\sin ky\right|}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]

Alternative 7: 48.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.05 \cdot 10^{-80}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin ky + 0.5 \cdot \frac{kx \cdot kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.05e-80)
   (* (sin th) (/ (sin ky) (+ (sin ky) (* 0.5 (/ (* kx kx) (sin ky))))))
   (* (sin ky) (/ (sin th) (hypot (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.05e-80) {
		tmp = sin(th) * (sin(ky) / (sin(ky) + (0.5 * ((kx * kx) / sin(ky)))));
	} else {
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.05e-80) {
		tmp = Math.sin(th) * (Math.sin(ky) / (Math.sin(ky) + (0.5 * ((kx * kx) / Math.sin(ky)))));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(Math.sin(ky), Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.05e-80:
		tmp = math.sin(th) * (math.sin(ky) / (math.sin(ky) + (0.5 * ((kx * kx) / math.sin(ky)))))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(math.sin(ky), math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.05e-80)
		tmp = Float64(sin(th) * Float64(sin(ky) / Float64(sin(ky) + Float64(0.5 * Float64(Float64(kx * kx) / sin(ky))))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(sin(ky), sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.05e-80)
		tmp = sin(th) * (sin(ky) / (sin(ky) + (0.5 * ((kx * kx) / sin(ky)))));
	else
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.05e-80], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] + N[(0.5 * N[(N[(kx * kx), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if kx < 1.05000000000000001e-80

   1. Initial program 56.0%

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

   2. Taylor expanded in kx around 0 45.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + 0.5 \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. unpow2 45.4%

         \[\leadsto \frac{\sin ky}{\sin ky + 0.5 \cdot \frac{\color{blue}{kx \cdot kx}}{\sin ky}} \cdot \sin th \]

   4. Simplified 45.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + 0.5 \cdot \frac{kx \cdot kx}{\sin ky}}} \cdot \sin th \]

   if 1.05000000000000001e-80 < kx

   1. Initial program 61.5%

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

      1. associate-*l/ 61.5%

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

      2. +-commutative 61.5%

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

      3. unpow2 61.5%

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

      4. unpow2 61.5%

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

      5. hypot-def 61.5%

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

   3. Simplified 61.5%

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

      1. associate-/l* 61.4%

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

      2. div-inv 61.4%

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

      3. clear-num 61.5%

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

   5. Applied egg-rr 61.5%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.05 \cdot 10^{-80}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin ky + 0.5 \cdot \frac{kx \cdot kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]

Alternative 8: 49.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \mathbf{if}\;ky \leq -0.029:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq -1.85 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -3 \cdot 10^{-274}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;ky \leq 3.8 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{ky}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin th}}\\ \mathbf{elif}\;ky \leq 1.76 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (/ (sin ky) (fabs (sin kx))))))
   (if (<= ky -0.029)
     (fabs (sin th))
     (if (<= ky -1.85e-79)
       t_1
       (if (<= ky -3e-274)
         (* (sin ky) (/ (sin th) (sin kx)))
         (if (<= ky 3.8e-162)
           t_1
           (if (<= ky 1.8e-87)
             (/ ky (/ (hypot (sin ky) kx) (sin th)))
             (if (<= ky 1.76e-16) t_1 (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / fabs(sin(kx)));
	double tmp;
	if (ky <= -0.029) {
		tmp = fabs(sin(th));
	} else if (ky <= -1.85e-79) {
		tmp = t_1;
	} else if (ky <= -3e-274) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (ky <= 3.8e-162) {
		tmp = t_1;
	} else if (ky <= 1.8e-87) {
		tmp = ky / (hypot(sin(ky), kx) / sin(th));
	} else if (ky <= 1.76e-16) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	double tmp;
	if (ky <= -0.029) {
		tmp = Math.abs(Math.sin(th));
	} else if (ky <= -1.85e-79) {
		tmp = t_1;
	} else if (ky <= -3e-274) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (ky <= 3.8e-162) {
		tmp = t_1;
	} else if (ky <= 1.8e-87) {
		tmp = ky / (Math.hypot(Math.sin(ky), kx) / Math.sin(th));
	} else if (ky <= 1.76e-16) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(kx)))
	tmp = 0
	if ky <= -0.029:
		tmp = math.fabs(math.sin(th))
	elif ky <= -1.85e-79:
		tmp = t_1
	elif ky <= -3e-274:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif ky <= 3.8e-162:
		tmp = t_1
	elif ky <= 1.8e-87:
		tmp = ky / (math.hypot(math.sin(ky), kx) / math.sin(th))
	elif ky <= 1.76e-16:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / abs(sin(kx))))
	tmp = 0.0
	if (ky <= -0.029)
		tmp = abs(sin(th));
	elseif (ky <= -1.85e-79)
		tmp = t_1;
	elseif (ky <= -3e-274)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (ky <= 3.8e-162)
		tmp = t_1;
	elseif (ky <= 1.8e-87)
		tmp = Float64(ky / Float64(hypot(sin(ky), kx) / sin(th)));
	elseif (ky <= 1.76e-16)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) * (sin(ky) / abs(sin(kx)));
	tmp = 0.0;
	if (ky <= -0.029)
		tmp = abs(sin(th));
	elseif (ky <= -1.85e-79)
		tmp = t_1;
	elseif (ky <= -3e-274)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (ky <= 3.8e-162)
		tmp = t_1;
	elseif (ky <= 1.8e-87)
		tmp = ky / (hypot(sin(ky), kx) / sin(th));
	elseif (ky <= 1.76e-16)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ky, -0.029], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[ky, -1.85e-79], t$95$1, If[LessEqual[ky, -3e-274], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 3.8e-162], t$95$1, If[LessEqual[ky, 1.8e-87], N[(ky / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 1.76e-16], t$95$1, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if ky < -0.0290000000000000015

   1. Initial program 50.2%

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

      1. associate-*l/ 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. unpow2 50.2%

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

      5. hypot-def 50.2%

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

   3. Simplified 50.2%

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

      1. add-sqr-sqrt 25.9%

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

      2. sqrt-unprod 33.3%

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

      3. pow2 33.3%

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

   5. Applied egg-rr 33.3%

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

      1. unpow2 33.3%

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

      2. rem-sqrt-square 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in kx around 0 56.7%

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

   if -0.0290000000000000015 < ky < -1.85000000000000009e-79 or -2.99999999999999977e-274 < ky < 3.80000000000000005e-162 or 1.79999999999999996e-87 < ky < 1.76e-16

   1. Initial program 78.3%

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

   2. Taylor expanded in ky around 0 48.2%

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

      1. add-sqr-sqrt 26.8%

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

      2. sqrt-prod 65.5%

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

      3. rem-sqrt-square 69.2%

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

   4. Applied egg-rr 69.2%

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

   if -1.85000000000000009e-79 < ky < -2.99999999999999977e-274

   1. Initial program 58.9%

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

      1. associate-*l/ 58.9%

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

      2. +-commutative 58.9%

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

      3. unpow2 58.9%

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

      4. unpow2 58.9%

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

      5. hypot-def 63.2%

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

   3. Simplified 63.2%

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

      1. associate-/l* 63.3%

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

      2. div-inv 63.2%

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

      3. clear-num 63.4%

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

   5. Applied egg-rr 63.4%

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

   6. Taylor expanded in ky around 0 61.8%

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

   if 3.80000000000000005e-162 < ky < 1.79999999999999996e-87

   1. Initial program 80.1%

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

      1. associate-/r/ 80.0%

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

      2. +-commutative 80.0%

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

      3. unpow2 80.0%

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

      4. sqr-neg 80.0%

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

      5. sin-neg 80.0%

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

      6. sin-neg 80.0%

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

      7. unpow2 80.0%

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

      8. +-commutative 80.0%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in kx around 0 70.9%

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

   5. Taylor expanded in ky around 0 70.9%

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

   if 1.76e-16 < ky

   1. Initial program 46.1%

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

      1. +-commutative 46.1%

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

      2. unpow2 46.1%

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

      3. unpow2 46.1%

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

      4. hypot-def 46.1%

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

   3. Simplified 46.1%

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

   4. Taylor expanded in kx around 0 53.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -0.029:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq -1.85 \cdot 10^{-79}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;ky \leq -3 \cdot 10^{-274}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;ky \leq 3.8 \cdot 10^{-162}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{ky}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin th}}\\ \mathbf{elif}\;ky \leq 1.76 \cdot 10^{-16}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9: 60.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -2.1:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 21000000:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -2.1)
   (fabs (sin th))
   (if (<= ky 21000000.0)
     (* (sin ky) (/ (sin th) (hypot ky (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -2.1) {
		tmp = fabs(sin(th));
	} else if (ky <= 21000000.0) {
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -2.1) {
		tmp = Math.abs(Math.sin(th));
	} else if (ky <= 21000000.0) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -2.1:
		tmp = math.fabs(math.sin(th))
	elif ky <= 21000000.0:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -2.1)
		tmp = abs(sin(th));
	elseif (ky <= 21000000.0)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(ky, sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -2.1)
		tmp = abs(sin(th));
	elseif (ky <= 21000000.0)
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -2.1], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[ky, 21000000.0], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < -2.10000000000000009

   1. Initial program 50.2%

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

      1. associate-*l/ 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. unpow2 50.2%

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

      5. hypot-def 50.2%

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

   3. Simplified 50.2%

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

      1. add-sqr-sqrt 25.9%

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

      2. sqrt-unprod 33.3%

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

      3. pow2 33.3%

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

   5. Applied egg-rr 33.3%

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

      1. unpow2 33.3%

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

      2. rem-sqrt-square 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in kx around 0 56.7%

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

   if -2.10000000000000009 < ky < 2.1e7

   1. Initial program 71.3%

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

      1. associate-*l/ 68.9%

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

      2. +-commutative 68.9%

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

      3. unpow2 68.9%

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

      4. unpow2 68.9%

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

      5. hypot-def 70.8%

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

   3. Simplified 70.8%

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

      1. associate-/l* 74.8%

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

      2. div-inv 74.7%

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

      3. clear-num 74.8%

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

   5. Applied egg-rr 74.8%

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

   6. Taylor expanded in ky around 0 73.0%

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

   if 2.1e7 < ky

   1. Initial program 44.1%

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

      1. +-commutative 44.1%

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

      2. unpow2 44.1%

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

      3. unpow2 44.1%

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

      4. hypot-def 44.1%

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

   3. Simplified 44.1%

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

   4. Taylor expanded in kx around 0 54.1%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -2.1:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 21000000:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 44.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;ky \leq -0.25:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -4 \cdot 10^{-70}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{t_1}}\\ \mathbf{elif}\;ky \leq -7 \cdot 10^{-74}:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;ky \leq 6.6 \cdot 10^{-155}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= ky -0.25)
     t_1
     (if (<= ky -4e-70)
       (/ ky (/ (sin kx) t_1))
       (if (<= ky -7e-74)
         (fabs th)
         (if (<= ky 6.6e-155) (* (sin ky) (/ (sin th) (sin kx))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (ky <= -0.25) {
		tmp = t_1;
	} else if (ky <= -4e-70) {
		tmp = ky / (sin(kx) / t_1);
	} else if (ky <= -7e-74) {
		tmp = fabs(th);
	} else if (ky <= 6.6e-155) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(sin(th))
    if (ky <= (-0.25d0)) then
        tmp = t_1
    else if (ky <= (-4d-70)) then
        tmp = ky / (sin(kx) / t_1)
    else if (ky <= (-7d-74)) then
        tmp = abs(th)
    else if (ky <= 6.6d-155) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double tmp;
	if (ky <= -0.25) {
		tmp = t_1;
	} else if (ky <= -4e-70) {
		tmp = ky / (Math.sin(kx) / t_1);
	} else if (ky <= -7e-74) {
		tmp = Math.abs(th);
	} else if (ky <= 6.6e-155) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	tmp = 0
	if ky <= -0.25:
		tmp = t_1
	elif ky <= -4e-70:
		tmp = ky / (math.sin(kx) / t_1)
	elif ky <= -7e-74:
		tmp = math.fabs(th)
	elif ky <= 6.6e-155:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	tmp = 0.0
	if (ky <= -0.25)
		tmp = t_1;
	elseif (ky <= -4e-70)
		tmp = Float64(ky / Float64(sin(kx) / t_1));
	elseif (ky <= -7e-74)
		tmp = abs(th);
	elseif (ky <= 6.6e-155)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	tmp = 0.0;
	if (ky <= -0.25)
		tmp = t_1;
	elseif (ky <= -4e-70)
		tmp = ky / (sin(kx) / t_1);
	elseif (ky <= -7e-74)
		tmp = abs(th);
	elseif (ky <= 6.6e-155)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ky, -0.25], t$95$1, If[LessEqual[ky, -4e-70], N[(ky / N[(N[Sin[kx], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, -7e-74], N[Abs[th], $MachinePrecision], If[LessEqual[ky, 6.6e-155], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if ky < -0.25

   1. Initial program 50.2%

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

      1. associate-*l/ 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. unpow2 50.2%

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

      5. hypot-def 50.2%

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

   3. Simplified 50.2%

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

      1. add-sqr-sqrt 25.9%

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

      2. sqrt-unprod 33.3%

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

      3. pow2 33.3%

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

   5. Applied egg-rr 33.3%

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

      1. unpow2 33.3%

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

      2. rem-sqrt-square 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in kx around 0 56.7%

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

   if -0.25 < ky < -3.99999999999999998e-70

   1. Initial program 83.3%

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

      1. associate-*l/ 82.9%

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

      2. +-commutative 82.9%

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

      3. unpow2 82.9%

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

      4. unpow2 82.9%

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

      5. hypot-def 82.9%

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

   3. Simplified 82.9%

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

      1. add-sqr-sqrt 74.2%

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

      2. sqrt-unprod 91.7%

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

      3. pow2 91.7%

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

   5. Applied egg-rr 91.7%

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

      1. unpow2 91.7%

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

      2. rem-sqrt-square 91.6%

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

   7. Simplified 91.6%

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

   8. Taylor expanded in ky around 0 18.8%

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

      1. associate-/l* 18.9%

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

   10. Simplified 18.9%

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

   if -3.99999999999999998e-70 < ky < -7.00000000000000029e-74

   1. Initial program 100.0%

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

      1. associate-*l/ 78.0%

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

      2. +-commutative 78.0%

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

      3. unpow2 78.0%

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

      4. unpow2 78.0%

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

      5. hypot-def 78.0%

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

   3. Simplified 78.0%

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

      1. add-sqr-sqrt 24.6%

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

      2. sqrt-unprod 31.0%

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

      3. pow2 31.0%

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

   5. Applied egg-rr 31.0%

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

      1. unpow2 31.0%

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

      2. rem-sqrt-square 30.2%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in th around 0 30.2%

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

   9. Taylor expanded in kx around 0 75.8%

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

   if -7.00000000000000029e-74 < ky < 6.59999999999999972e-155

   1. Initial program 65.1%

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

      1. associate-*l/ 65.1%

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

      2. +-commutative 65.1%

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

      3. unpow2 65.1%

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

      4. unpow2 65.1%

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

      5. hypot-def 68.0%

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

   3. Simplified 68.0%

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

      1. associate-/l* 70.6%

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

      2. div-inv 70.5%

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

      3. clear-num 70.6%

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

   5. Applied egg-rr 70.6%

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

   6. Taylor expanded in ky around 0 60.7%

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

   if 6.59999999999999972e-155 < ky

   1. Initial program 52.2%

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

      1. +-commutative 52.2%

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

      2. unpow2 52.2%

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

      3. unpow2 52.2%

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

      4. hypot-def 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in kx around 0 50.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -0.25:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq -4 \cdot 10^{-70}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\left|\sin th\right|}}\\ \mathbf{elif}\;ky \leq -7 \cdot 10^{-74}:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;ky \leq 6.6 \cdot 10^{-155}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 44.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;ky \leq -0.03:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -3.7 \cdot 10^{-66}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{t_1}}\\ \mathbf{elif}\;ky \leq -6 \cdot 10^{-74}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= ky -0.03)
     t_1
     (if (<= ky -3.7e-66)
       (/ ky (/ (sin kx) t_1))
       (if (<= ky -6e-74)
         (* th (/ (sin ky) (hypot (sin ky) kx)))
         (if (<= ky 7.5e-155) (* (sin ky) (/ (sin th) (sin kx))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (ky <= -0.03) {
		tmp = t_1;
	} else if (ky <= -3.7e-66) {
		tmp = ky / (sin(kx) / t_1);
	} else if (ky <= -6e-74) {
		tmp = th * (sin(ky) / hypot(sin(ky), kx));
	} else if (ky <= 7.5e-155) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double tmp;
	if (ky <= -0.03) {
		tmp = t_1;
	} else if (ky <= -3.7e-66) {
		tmp = ky / (Math.sin(kx) / t_1);
	} else if (ky <= -6e-74) {
		tmp = th * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else if (ky <= 7.5e-155) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	tmp = 0
	if ky <= -0.03:
		tmp = t_1
	elif ky <= -3.7e-66:
		tmp = ky / (math.sin(kx) / t_1)
	elif ky <= -6e-74:
		tmp = th * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	elif ky <= 7.5e-155:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	tmp = 0.0
	if (ky <= -0.03)
		tmp = t_1;
	elseif (ky <= -3.7e-66)
		tmp = Float64(ky / Float64(sin(kx) / t_1));
	elseif (ky <= -6e-74)
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (ky <= 7.5e-155)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	tmp = 0.0;
	if (ky <= -0.03)
		tmp = t_1;
	elseif (ky <= -3.7e-66)
		tmp = ky / (sin(kx) / t_1);
	elseif (ky <= -6e-74)
		tmp = th * (sin(ky) / hypot(sin(ky), kx));
	elseif (ky <= 7.5e-155)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ky, -0.03], t$95$1, If[LessEqual[ky, -3.7e-66], N[(ky / N[(N[Sin[kx], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, -6e-74], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 7.5e-155], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if ky < -0.029999999999999999

   1. Initial program 50.2%

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

      1. associate-*l/ 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. unpow2 50.2%

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

      5. hypot-def 50.2%

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

   3. Simplified 50.2%

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

      1. add-sqr-sqrt 25.9%

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

      2. sqrt-unprod 33.3%

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

      3. pow2 33.3%

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

   5. Applied egg-rr 33.3%

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

      1. unpow2 33.3%

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

      2. rem-sqrt-square 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in kx around 0 56.7%

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

   if -0.029999999999999999 < ky < -3.7000000000000002e-66

   1. Initial program 83.3%

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

      1. associate-*l/ 82.9%

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

      2. +-commutative 82.9%

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

      3. unpow2 82.9%

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

      4. unpow2 82.9%

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

      5. hypot-def 82.9%

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

   3. Simplified 82.9%

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

      1. add-sqr-sqrt 74.2%

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

      2. sqrt-unprod 91.7%

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

      3. pow2 91.7%

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

   5. Applied egg-rr 91.7%

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

      1. unpow2 91.7%

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

      2. rem-sqrt-square 91.6%

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

   7. Simplified 91.6%

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

   8. Taylor expanded in ky around 0 18.8%

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

      1. associate-/l* 18.9%

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

   10. Simplified 18.9%

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

   if -3.7000000000000002e-66 < ky < -6.00000000000000014e-74

   1. Initial program 100.0%

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

      1. associate-/r/ 98.8%

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

      2. +-commutative 98.8%

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

      3. unpow2 98.8%

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

      4. sqr-neg 98.8%

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

      5. sin-neg 98.8%

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

      6. sin-neg 98.8%

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

      7. unpow2 98.8%

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

      8. +-commutative 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in kx around 0 75.6%

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

   5. Taylor expanded in th around 0 75.6%

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

      1. associate-*l/ 75.6%

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

      2. +-commutative 75.6%

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

      3. unpow2 75.6%

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

      4. unpow2 75.6%

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

      5. hypot-def 75.6%

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

      6. *-lft-identity 75.6%

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

   7. Simplified 75.6%

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

      1. associate-/r/ 76.3%

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

   9. Applied egg-rr 76.3%

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

   if -6.00000000000000014e-74 < ky < 7.5000000000000006e-155

   1. Initial program 65.1%

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

      1. associate-*l/ 65.1%

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

      2. +-commutative 65.1%

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

      3. unpow2 65.1%

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

      4. unpow2 65.1%

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

      5. hypot-def 68.0%

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

   3. Simplified 68.0%

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

      1. associate-/l* 70.6%

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

      2. div-inv 70.5%

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

      3. clear-num 70.6%

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

   5. Applied egg-rr 70.6%

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

   6. Taylor expanded in ky around 0 60.7%

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

   if 7.5000000000000006e-155 < ky

   1. Initial program 52.2%

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

      1. +-commutative 52.2%

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

      2. unpow2 52.2%

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

      3. unpow2 52.2%

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

      4. hypot-def 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in kx around 0 50.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -0.03:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq -3.7 \cdot 10^{-66}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\left|\sin th\right|}}\\ \mathbf{elif}\;ky \leq -6 \cdot 10^{-74}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 44.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -5.5e-5)
   (fabs (sin th))
   (if (<= ky 7.5e-155) (* (sin ky) (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5.5e-5) {
		tmp = fabs(sin(th));
	} else if (ky <= 7.5e-155) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-5.5d-5)) then
        tmp = abs(sin(th))
    else if (ky <= 7.5d-155) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5.5e-5) {
		tmp = Math.abs(Math.sin(th));
	} else if (ky <= 7.5e-155) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -5.5e-5:
		tmp = math.fabs(math.sin(th))
	elif ky <= 7.5e-155:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -5.5e-5)
		tmp = abs(sin(th));
	elseif (ky <= 7.5e-155)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -5.5e-5)
		tmp = abs(sin(th));
	elseif (ky <= 7.5e-155)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -5.5e-5], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[ky, 7.5e-155], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < -5.5000000000000002e-5

   1. Initial program 51.7%

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

      1. associate-*l/ 51.8%

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

      2. +-commutative 51.8%

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

      3. unpow2 51.8%

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

      4. unpow2 51.8%

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

      5. hypot-def 51.8%

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

   3. Simplified 51.8%

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

      1. add-sqr-sqrt 26.6%

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

      2. sqrt-unprod 34.0%

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

      3. pow2 34.0%

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

   5. Applied egg-rr 34.0%

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

      1. unpow2 34.0%

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

      2. rem-sqrt-square 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in kx around 0 55.1%

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

   if -5.5000000000000002e-5 < ky < 7.5000000000000006e-155

   1. Initial program 68.5%

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

      1. associate-*l/ 67.4%

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

      2. +-commutative 67.4%

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

      3. unpow2 67.4%

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

      4. unpow2 67.4%

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

      5. hypot-def 69.8%

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

   3. Simplified 69.8%

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

      1. associate-/l* 73.0%

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

      2. div-inv 72.9%

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

      3. clear-num 73.0%

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

   5. Applied egg-rr 73.0%

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

   6. Taylor expanded in ky around 0 54.7%

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

   if 7.5000000000000006e-155 < ky

   1. Initial program 52.2%

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

      1. +-commutative 52.2%

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

      2. unpow2 52.2%

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

      3. unpow2 52.2%

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

      4. hypot-def 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in kx around 0 50.4%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 44.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -8 \cdot 10^{-5}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -8e-5)
   (fabs (sin th))
   (if (<= ky 7.5e-155) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -8e-5) {
		tmp = fabs(sin(th));
	} else if (ky <= 7.5e-155) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-8d-5)) then
        tmp = abs(sin(th))
    else if (ky <= 7.5d-155) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -8e-5) {
		tmp = Math.abs(Math.sin(th));
	} else if (ky <= 7.5e-155) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -8e-5:
		tmp = math.fabs(math.sin(th))
	elif ky <= 7.5e-155:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -8e-5)
		tmp = abs(sin(th));
	elseif (ky <= 7.5e-155)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -8e-5)
		tmp = abs(sin(th));
	elseif (ky <= 7.5e-155)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -8e-5], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[ky, 7.5e-155], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < -8.00000000000000065e-5

   1. Initial program 51.7%

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

      1. associate-*l/ 51.8%

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

      2. +-commutative 51.8%

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

      3. unpow2 51.8%

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

      4. unpow2 51.8%

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

      5. hypot-def 51.8%

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

   3. Simplified 51.8%

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

      1. add-sqr-sqrt 26.6%

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

      2. sqrt-unprod 34.0%

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

      3. pow2 34.0%

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

   5. Applied egg-rr 34.0%

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

      1. unpow2 34.0%

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

      2. rem-sqrt-square 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in kx around 0 55.1%

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

   if -8.00000000000000065e-5 < ky < 7.5000000000000006e-155

   1. Initial program 68.5%

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

   2. Taylor expanded in ky around 0 54.6%

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

   3. Taylor expanded in ky around 0 54.6%

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

   if 7.5000000000000006e-155 < ky

   1. Initial program 52.2%

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

      1. +-commutative 52.2%

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

      2. unpow2 52.2%

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

      3. unpow2 52.2%

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

      4. hypot-def 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in kx around 0 50.4%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -8 \cdot 10^{-5}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 44.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -8 \cdot 10^{-5}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -8e-5)
   (fabs (sin th))
   (if (<= ky 7.5e-155) (/ ky (/ (sin kx) (sin th))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -8e-5) {
		tmp = fabs(sin(th));
	} else if (ky <= 7.5e-155) {
		tmp = ky / (sin(kx) / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-8d-5)) then
        tmp = abs(sin(th))
    else if (ky <= 7.5d-155) then
        tmp = ky / (sin(kx) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -8e-5) {
		tmp = Math.abs(Math.sin(th));
	} else if (ky <= 7.5e-155) {
		tmp = ky / (Math.sin(kx) / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -8e-5:
		tmp = math.fabs(math.sin(th))
	elif ky <= 7.5e-155:
		tmp = ky / (math.sin(kx) / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -8e-5)
		tmp = abs(sin(th));
	elseif (ky <= 7.5e-155)
		tmp = Float64(ky / Float64(sin(kx) / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -8e-5)
		tmp = abs(sin(th));
	elseif (ky <= 7.5e-155)
		tmp = ky / (sin(kx) / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -8e-5], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[ky, 7.5e-155], N[(ky / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < -8.00000000000000065e-5

   1. Initial program 51.7%

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

      1. associate-*l/ 51.8%

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

      2. +-commutative 51.8%

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

      3. unpow2 51.8%

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

      4. unpow2 51.8%

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

      5. hypot-def 51.8%

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

   3. Simplified 51.8%

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

      1. add-sqr-sqrt 26.6%

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

      2. sqrt-unprod 34.0%

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

      3. pow2 34.0%

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

   5. Applied egg-rr 34.0%

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

      1. unpow2 34.0%

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

      2. rem-sqrt-square 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in kx around 0 55.1%

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

   if -8.00000000000000065e-5 < ky < 7.5000000000000006e-155

   1. Initial program 68.5%

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

      1. +-commutative 68.5%

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

      2. unpow2 68.5%

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

      3. unpow2 68.5%

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

      4. hypot-def 73.0%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in ky around 0 53.6%

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

      1. associate-/l* 54.6%

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

   6. Simplified 54.6%

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

   if 7.5000000000000006e-155 < ky

   1. Initial program 52.2%

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

      1. +-commutative 52.2%

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

      2. unpow2 52.2%

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

      3. unpow2 52.2%

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

      4. hypot-def 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in kx around 0 50.4%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -8 \cdot 10^{-5}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15: 37.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -5.7 \cdot 10^{-5}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 2.5 \cdot 10^{-176}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -5.7e-5)
   (fabs (sin th))
   (if (<= ky 2.5e-176) (* ky (/ th (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5.7e-5) {
		tmp = fabs(sin(th));
	} else if (ky <= 2.5e-176) {
		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 <= (-5.7d-5)) then
        tmp = abs(sin(th))
    else if (ky <= 2.5d-176) 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 <= -5.7e-5) {
		tmp = Math.abs(Math.sin(th));
	} else if (ky <= 2.5e-176) {
		tmp = ky * (th / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -5.7e-5:
		tmp = math.fabs(math.sin(th))
	elif ky <= 2.5e-176:
		tmp = ky * (th / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -5.7e-5)
		tmp = abs(sin(th));
	elseif (ky <= 2.5e-176)
		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 <= -5.7e-5)
		tmp = abs(sin(th));
	elseif (ky <= 2.5e-176)
		tmp = ky * (th / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -5.7e-5], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[ky, 2.5e-176], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < -5.7000000000000003e-5

   1. Initial program 51.7%

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

      1. associate-*l/ 51.8%

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

      2. +-commutative 51.8%

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

      3. unpow2 51.8%

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

      4. unpow2 51.8%

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

      5. hypot-def 51.8%

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

   3. Simplified 51.8%

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

      1. add-sqr-sqrt 26.6%

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

      2. sqrt-unprod 34.0%

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

      3. pow2 34.0%

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

   5. Applied egg-rr 34.0%

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

      1. unpow2 34.0%

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

      2. rem-sqrt-square 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in kx around 0 55.1%

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

   if -5.7000000000000003e-5 < ky < 2.5e-176

   1. Initial program 68.5%

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

      1. associate-*l/ 67.4%

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

      2. +-commutative 67.4%

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

      3. unpow2 67.4%

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

      4. unpow2 67.4%

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

      5. hypot-def 69.9%

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

   3. Simplified 69.9%

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

      1. add-sqr-sqrt 50.7%

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

      2. sqrt-unprod 85.4%

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

      3. pow2 85.4%

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

   5. Applied egg-rr 85.4%

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

      1. unpow2 85.4%

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

      2. rem-sqrt-square 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in th around 0 33.4%

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

   9. Taylor expanded in ky around 0 26.0%

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

       1. associate-/l* 26.9%

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

   11. Simplified 26.9%

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

       1. clear-num 26.9%

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

       2. associate-/r/ 26.9%

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

       3. clear-num 26.9%

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

       4. remove-double-div 26.9%

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

       5. add-sqr-sqrt 15.6%

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

       6. fabs-sqr 15.6%

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

       7. add-sqr-sqrt 29.9%

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

   13. Applied egg-rr 29.9%

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

   if 2.5e-176 < ky

   1. Initial program 52.6%

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

      1. +-commutative 52.6%

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

      2. unpow2 52.6%

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

      3. unpow2 52.6%

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

      4. hypot-def 52.6%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in kx around 0 49.0%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -5.7 \cdot 10^{-5}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 2.5 \cdot 10^{-176}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16: 38.1% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.14 \cdot 10^{-41}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.6 \cdot 10^{-176}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.14e-41)
   (sin th)
   (if (<= ky 2.6e-176) (* ky (/ th (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.14e-41) {
		tmp = sin(th);
	} else if (ky <= 2.6e-176) {
		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 <= (-1.14d-41)) then
        tmp = sin(th)
    else if (ky <= 2.6d-176) 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 <= -1.14e-41) {
		tmp = Math.sin(th);
	} else if (ky <= 2.6e-176) {
		tmp = ky * (th / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.14e-41:
		tmp = math.sin(th)
	elif ky <= 2.6e-176:
		tmp = ky * (th / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.14e-41)
		tmp = sin(th);
	elseif (ky <= 2.6e-176)
		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 <= -1.14e-41)
		tmp = sin(th);
	elseif (ky <= 2.6e-176)
		tmp = ky * (th / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.14e-41], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 2.6e-176], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.14e-41 or 2.59999999999999992e-176 < ky

   1. Initial program 52.5%

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

      1. +-commutative 52.5%

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

      2. unpow2 52.5%

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

      3. unpow2 52.5%

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

      4. hypot-def 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in kx around 0 48.1%

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

   if -1.14e-41 < ky < 2.59999999999999992e-176

   1. Initial program 69.0%

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

      1. associate-*l/ 67.9%

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

      2. +-commutative 67.9%

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

      3. unpow2 67.9%

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

      4. unpow2 67.9%

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

      5. hypot-def 70.6%

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

   3. Simplified 70.6%

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

      1. add-sqr-sqrt 50.2%

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

      2. sqrt-unprod 84.5%

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

      3. pow2 84.5%

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

   5. Applied egg-rr 84.5%

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

      1. unpow2 84.5%

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

      2. rem-sqrt-square 84.4%

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

   7. Simplified 84.4%

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

   8. Taylor expanded in th around 0 35.2%

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

   9. Taylor expanded in ky around 0 27.5%

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

       1. associate-/l* 28.5%

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

   11. Simplified 28.5%

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

       1. clear-num 28.5%

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

       2. associate-/r/ 28.5%

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

       3. clear-num 28.5%

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

       4. remove-double-div 28.5%

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

       5. add-sqr-sqrt 16.6%

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

       6. fabs-sqr 16.6%

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

       7. add-sqr-sqrt 31.7%

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

   13. Applied egg-rr 31.7%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.14 \cdot 10^{-41}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.6 \cdot 10^{-176}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17: 37.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.15 \cdot 10^{-36}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.95 \cdot 10^{-157}:\\ \;\;\;\;\frac{ky}{0.16666666666666666 \cdot \left(th \cdot kx\right) + \frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.15e-36)
   (sin th)
   (if (<= ky 2.95e-157)
     (/ ky (+ (* 0.16666666666666666 (* th kx)) (/ kx th)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.15e-36) {
		tmp = sin(th);
	} else if (ky <= 2.95e-157) {
		tmp = ky / ((0.16666666666666666 * (th * kx)) + (kx / th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-1.15d-36)) then
        tmp = sin(th)
    else if (ky <= 2.95d-157) then
        tmp = ky / ((0.16666666666666666d0 * (th * kx)) + (kx / th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.15e-36) {
		tmp = Math.sin(th);
	} else if (ky <= 2.95e-157) {
		tmp = ky / ((0.16666666666666666 * (th * kx)) + (kx / th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.15e-36:
		tmp = math.sin(th)
	elif ky <= 2.95e-157:
		tmp = ky / ((0.16666666666666666 * (th * kx)) + (kx / th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.15e-36)
		tmp = sin(th);
	elseif (ky <= 2.95e-157)
		tmp = Float64(ky / Float64(Float64(0.16666666666666666 * Float64(th * kx)) + Float64(kx / th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.15e-36)
		tmp = sin(th);
	elseif (ky <= 2.95e-157)
		tmp = ky / ((0.16666666666666666 * (th * kx)) + (kx / th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.15e-36], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 2.95e-157], N[(ky / N[(N[(0.16666666666666666 * N[(th * kx), $MachinePrecision]), $MachinePrecision] + N[(kx / th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.14999999999999998e-36 or 2.9500000000000001e-157 < ky

   1. Initial program 52.0%

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

      1. +-commutative 52.0%

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

      2. unpow2 52.0%

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

      3. unpow2 52.0%

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

      4. hypot-def 52.0%

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

   3. Simplified 52.0%

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

   4. Taylor expanded in kx around 0 49.2%

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

   if -1.14999999999999998e-36 < ky < 2.9500000000000001e-157

   1. Initial program 69.3%

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

      1. associate-/r/ 69.3%

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

      2. +-commutative 69.3%

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

      3. unpow2 69.3%

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

      4. sqr-neg 69.3%

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

      5. sin-neg 69.3%

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

      6. sin-neg 69.3%

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

      7. unpow2 69.3%

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

      8. +-commutative 69.3%

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

   3. Simplified 74.1%

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

   4. Taylor expanded in kx around 0 45.1%

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

   5. Taylor expanded in ky around 0 28.7%

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

      1. associate-/l* 29.6%

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

   7. Simplified 29.6%

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

   8. Taylor expanded in th around 0 25.8%

      \[\leadsto \frac{ky}{\color{blue}{0.16666666666666666 \cdot \left(kx \cdot th\right) + \frac{kx}{th}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.15 \cdot 10^{-36}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.95 \cdot 10^{-157}:\\ \;\;\;\;\frac{ky}{0.16666666666666666 \cdot \left(th \cdot kx\right) + \frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 18: 27.8% accurate, 46.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -0.82:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{ky}{0.16666666666666666 \cdot \left(th \cdot kx\right) + \frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -0.82)
   th
   (if (<= ky 7.5e-155)
     (/ ky (+ (* 0.16666666666666666 (* th kx)) (/ kx th)))
     th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -0.82) {
		tmp = th;
	} else if (ky <= 7.5e-155) {
		tmp = ky / ((0.16666666666666666 * (th * kx)) + (kx / th));
	} else {
		tmp = 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 <= (-0.82d0)) then
        tmp = th
    else if (ky <= 7.5d-155) then
        tmp = ky / ((0.16666666666666666d0 * (th * kx)) + (kx / th))
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -0.82) {
		tmp = th;
	} else if (ky <= 7.5e-155) {
		tmp = ky / ((0.16666666666666666 * (th * kx)) + (kx / th));
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -0.82:
		tmp = th
	elif ky <= 7.5e-155:
		tmp = ky / ((0.16666666666666666 * (th * kx)) + (kx / th))
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -0.82)
		tmp = th;
	elseif (ky <= 7.5e-155)
		tmp = Float64(ky / Float64(Float64(0.16666666666666666 * Float64(th * kx)) + Float64(kx / th)));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -0.82)
		tmp = th;
	elseif (ky <= 7.5e-155)
		tmp = ky / ((0.16666666666666666 * (th * kx)) + (kx / th));
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -0.82], th, If[LessEqual[ky, 7.5e-155], N[(ky / N[(N[(0.16666666666666666 * N[(th * kx), $MachinePrecision]), $MachinePrecision] + N[(kx / th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -0.819999999999999951 or 7.5000000000000006e-155 < ky

   1. Initial program 51.5%

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

      1. +-commutative 51.5%

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

      2. unpow2 51.5%

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

      3. unpow2 51.5%

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

      4. hypot-def 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in kx around 0 50.2%

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

   5. Taylor expanded in th around 0 35.7%

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

   if -0.819999999999999951 < ky < 7.5000000000000006e-155

   1. Initial program 69.2%

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

      1. associate-/r/ 69.2%

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

      2. +-commutative 69.2%

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

      3. unpow2 69.2%

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

      4. sqr-neg 69.2%

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

      5. sin-neg 69.2%

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

      6. sin-neg 69.2%

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

      7. unpow2 69.2%

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

      8. +-commutative 69.2%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in kx around 0 44.6%

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

   5. Taylor expanded in ky around 0 26.9%

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

      1. associate-/l* 27.8%

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

   7. Simplified 27.8%

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

   8. Taylor expanded in th around 0 24.3%

      \[\leadsto \frac{ky}{\color{blue}{0.16666666666666666 \cdot \left(kx \cdot th\right) + \frac{kx}{th}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -0.82:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{ky}{0.16666666666666666 \cdot \left(th \cdot kx\right) + \frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 19: 27.7% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -5.5 \cdot 10^{-28}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 7.3 \cdot 10^{-155}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -5.5e-28) th (if (<= ky 7.3e-155) (* th (/ ky kx)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5.5e-28) {
		tmp = th;
	} else if (ky <= 7.3e-155) {
		tmp = th * (ky / kx);
	} else {
		tmp = 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 <= (-5.5d-28)) then
        tmp = th
    else if (ky <= 7.3d-155) then
        tmp = th * (ky / kx)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5.5e-28) {
		tmp = th;
	} else if (ky <= 7.3e-155) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -5.5e-28:
		tmp = th
	elif ky <= 7.3e-155:
		tmp = th * (ky / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -5.5e-28)
		tmp = th;
	elseif (ky <= 7.3e-155)
		tmp = Float64(th * Float64(ky / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -5.5e-28)
		tmp = th;
	elseif (ky <= 7.3e-155)
		tmp = th * (ky / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -5.5e-28], th, If[LessEqual[ky, 7.3e-155], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -5.49999999999999967e-28 or 7.30000000000000033e-155 < ky

   1. Initial program 52.0%

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

      1. +-commutative 52.0%

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

      2. unpow2 52.0%

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

      3. unpow2 52.0%

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

      4. hypot-def 52.0%

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

   3. Simplified 52.0%

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

   4. Taylor expanded in kx around 0 49.2%

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

   5. Taylor expanded in th around 0 34.6%

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

   if -5.49999999999999967e-28 < ky < 7.30000000000000033e-155

   1. Initial program 69.3%

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

      1. associate-/r/ 69.3%

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

      2. +-commutative 69.3%

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

      3. unpow2 69.3%

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

      4. sqr-neg 69.3%

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

      5. sin-neg 69.3%

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

      6. sin-neg 69.3%

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

      7. unpow2 69.3%

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

      8. +-commutative 69.3%

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

   3. Simplified 74.1%

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

   4. Taylor expanded in kx around 0 45.1%

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

   5. Taylor expanded in ky around 0 28.7%

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

      1. associate-/l* 29.6%

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

   7. Simplified 29.6%

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

   8. Taylor expanded in th around 0 24.6%

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

      1. associate-/l* 25.6%

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

   10. Simplified 25.6%

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

       1. associate-/r/ 25.5%

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

   12. Applied egg-rr 25.5%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -5.5 \cdot 10^{-28}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 7.3 \cdot 10^{-155}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 20: 19.6% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 57.6%

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

   1. +-commutative 57.6%

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

   2. unpow2 57.6%

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

   3. unpow2 57.6%

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

   4. hypot-def 59.2%

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

3. Simplified 59.2%

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

4. Taylor expanded in kx around 0 34.6%

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

5. Taylor expanded in th around 0 24.5%

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

6. Final simplification 24.5%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023292 
(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)))