Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 12.9s

Alternatives: 21

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

   1. unpow2 95.1%

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

   2. sqr-neg 95.1%

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

   3. sin-neg 95.1%

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

   4. sin-neg 95.1%

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

   5. unpow2 95.1%

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

   6. associate-*l/ 93.3%

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

   7. associate-/l* 95.1%

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

   8. +-commutative 95.1%

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

   9. unpow2 95.1%

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

   10. sin-neg 95.1%

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

   11. sin-neg 95.1%

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

3. Simplified 99.6%

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

   1. associate-*r/ 97.1%

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

   2. hypot-undefine 93.3%

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

   3. unpow2 93.3%

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

   4. unpow2 93.3%

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

   5. +-commutative 93.3%

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

   6. associate-*l/ 95.1%

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

   7. *-commutative 95.1%

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

   8. clear-num 95.1%

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

   9. un-div-inv 95.1%

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

   10. +-commutative 95.1%

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

   11. unpow2 95.1%

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

   12. unpow2 95.1%

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

   13. hypot-undefine 99.7%

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

6. Applied egg-rr 99.7%

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

Alternative 2: 58.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.0005:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq 0.002:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.0005)
   (/ (* th (sin ky)) (hypot (sin ky) kx))
   (if (<= (sin ky) 0.002) (/ (* (sin th) ky) (fabs (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.0005) {
		tmp = (th * sin(ky)) / hypot(sin(ky), kx);
	} else if (sin(ky) <= 0.002) {
		tmp = (sin(th) * ky) / fabs(sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.0005) {
		tmp = (th * Math.sin(ky)) / Math.hypot(Math.sin(ky), kx);
	} else if (Math.sin(ky) <= 0.002) {
		tmp = (Math.sin(th) * ky) / Math.abs(Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.0005:
		tmp = (th * math.sin(ky)) / math.hypot(math.sin(ky), kx)
	elif math.sin(ky) <= 0.002:
		tmp = (math.sin(th) * ky) / math.fabs(math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.0005)
		tmp = Float64(Float64(th * sin(ky)) / hypot(sin(ky), kx));
	elseif (sin(ky) <= 0.002)
		tmp = Float64(Float64(sin(th) * ky) / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.0005)
		tmp = (th * sin(ky)) / hypot(sin(ky), kx);
	elseif (sin(ky) <= 0.002)
		tmp = (sin(th) * ky) / abs(sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.0005], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.002], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -5.0000000000000001e-4

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in th around 0 58.1%

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

      1. rem-square-sqrt 58.1%

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

      2. +-commutative 58.1%

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

      3. unpow2 58.1%

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

      4. unpow2 58.1%

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

      5. hypot-undefine 58.1%

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

      6. +-commutative 58.1%

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

      7. unpow2 58.1%

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

      8. unpow2 58.1%

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

      9. hypot-undefine 58.1%

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

      10. unpow2 58.1%

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

   7. Simplified 58.1%

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

      1. associate-*r* 58.3%

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

      2. sqrt-div 58.2%

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

      3. metadata-eval 58.2%

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

      4. sqrt-pow1 58.2%

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

      5. metadata-eval 58.2%

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

      6. pow1 58.2%

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

      7. un-div-inv 58.3%

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

      8. *-commutative 58.3%

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

      9. hypot-undefine 58.3%

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

      10. +-commutative 58.3%

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

      11. hypot-undefine 58.3%

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

   9. Applied egg-rr 58.3%

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

   10. Taylor expanded in kx around 0 29.7%

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

   if -5.0000000000000001e-4 < (sin.f64 ky) < 2e-3

   1. Initial program 90.2%

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

      1. unpow2 90.2%

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

      2. sqr-neg 90.2%

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

      3. sin-neg 90.2%

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

      4. sin-neg 90.2%

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

      5. unpow2 90.2%

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

      6. associate-*l/ 86.6%

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

      7. associate-/l* 90.3%

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

      8. +-commutative 90.3%

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

      9. unpow2 90.3%

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

      10. sin-neg 90.3%

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

      11. sin-neg 90.3%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 45.4%

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

      1. add-sqr-sqrt 38.2%

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

      2. sqrt-prod 75.2%

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

      3. rem-sqrt-square 81.9%

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

   7. Applied egg-rr 78.9%

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

   if 2e-3 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 59.7%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.0005:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq 0.002:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.0005:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq 0.002:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.0005)
   (* (sin ky) (/ th (hypot (sin ky) kx)))
   (if (<= (sin ky) 0.002) (/ (* (sin th) ky) (fabs (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.0005) {
		tmp = sin(ky) * (th / hypot(sin(ky), kx));
	} else if (sin(ky) <= 0.002) {
		tmp = (sin(th) * ky) / fabs(sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.0005) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), kx));
	} else if (Math.sin(ky) <= 0.002) {
		tmp = (Math.sin(th) * ky) / Math.abs(Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.0005:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), kx))
	elif math.sin(ky) <= 0.002:
		tmp = (math.sin(th) * ky) / math.fabs(math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.0005)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), kx)));
	elseif (sin(ky) <= 0.002)
		tmp = Float64(Float64(sin(th) * ky) / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.0005)
		tmp = sin(ky) * (th / hypot(sin(ky), kx));
	elseif (sin(ky) <= 0.002)
		tmp = (sin(th) * ky) / abs(sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.0005], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.002], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -5.0000000000000001e-4

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 46.7%

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

   6. Taylor expanded in th around 0 29.7%

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

   if -5.0000000000000001e-4 < (sin.f64 ky) < 2e-3

   1. Initial program 90.2%

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

      1. unpow2 90.2%

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

      2. sqr-neg 90.2%

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

      3. sin-neg 90.2%

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

      4. sin-neg 90.2%

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

      5. unpow2 90.2%

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

      6. associate-*l/ 86.6%

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

      7. associate-/l* 90.3%

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

      8. +-commutative 90.3%

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

      9. unpow2 90.3%

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

      10. sin-neg 90.3%

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

      11. sin-neg 90.3%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 45.4%

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

      1. add-sqr-sqrt 38.2%

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

      2. sqrt-prod 75.2%

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

      3. rem-sqrt-square 81.9%

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

   7. Applied egg-rr 78.9%

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

   if 2e-3 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 59.7%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.0005:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq 0.002:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.0005:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 0.002:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.0005)
   (sqrt (pow (sin th) 2.0))
   (if (<= (sin ky) 0.002) (/ (* (sin th) ky) (fabs (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.0005) {
		tmp = sqrt(pow(sin(th), 2.0));
	} else if (sin(ky) <= 0.002) {
		tmp = (sin(th) * ky) / fabs(sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.0005d0)) then
        tmp = sqrt((sin(th) ** 2.0d0))
    else if (sin(ky) <= 0.002d0) then
        tmp = (sin(th) * ky) / abs(sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.0005) {
		tmp = Math.sqrt(Math.pow(Math.sin(th), 2.0));
	} else if (Math.sin(ky) <= 0.002) {
		tmp = (Math.sin(th) * ky) / Math.abs(Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.0005:
		tmp = math.sqrt(math.pow(math.sin(th), 2.0))
	elif math.sin(ky) <= 0.002:
		tmp = (math.sin(th) * ky) / math.fabs(math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.0005)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 0.002)
		tmp = Float64(Float64(sin(th) * ky) / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.0005)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 0.002)
		tmp = (sin(th) * ky) / abs(sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.0005], N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.002], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -5.0000000000000001e-4

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 2.9%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 26.5%

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

      3. pow2 26.5%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   7. Applied egg-rr 26.5%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]

   if -5.0000000000000001e-4 < (sin.f64 ky) < 2e-3

   1. Initial program 90.2%

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

      1. unpow2 90.2%

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

      2. sqr-neg 90.2%

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

      3. sin-neg 90.2%

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

      4. sin-neg 90.2%

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

      5. unpow2 90.2%

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

      6. associate-*l/ 86.6%

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

      7. associate-/l* 90.3%

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

      8. +-commutative 90.3%

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

      9. unpow2 90.3%

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

      10. sin-neg 90.3%

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

      11. sin-neg 90.3%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 45.4%

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

      1. add-sqr-sqrt 38.2%

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

      2. sqrt-prod 75.2%

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

      3. rem-sqrt-square 81.9%

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

   7. Applied egg-rr 78.9%

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

   if 2e-3 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 59.7%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.0005:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 0.002:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-119}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.02)
   (fabs (* ky (/ (sin th) (sin kx))))
   (if (<= (sin kx) 2e-119) (sin th) (/ (sin th) (/ (sin kx) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.02) {
		tmp = fabs((ky * (sin(th) / sin(kx))));
	} else if (sin(kx) <= 2e-119) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(kx) / sin(ky));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.02d0)) then
        tmp = abs((ky * (sin(th) / sin(kx))))
    else if (sin(kx) <= 2d-119) then
        tmp = sin(th)
    else
        tmp = sin(th) / (sin(kx) / sin(ky))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.02) {
		tmp = Math.abs((ky * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(kx) <= 2e-119) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.02:
		tmp = math.fabs((ky * (math.sin(th) / math.sin(kx))))
	elif math.sin(kx) <= 2e-119:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.02)
		tmp = abs(Float64(ky * Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= 2e-119)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.02)
		tmp = abs((ky * (sin(th) / sin(kx))));
	elseif (sin(kx) <= 2e-119)
		tmp = sin(th);
	else
		tmp = sin(th) / (sin(kx) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], N[Abs[N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-119], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0200000000000000004

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 14.7%

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

      1. add-sqr-sqrt 12.0%

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

      2. sqrt-unprod 25.6%

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

      3. pow2 25.6%

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

      4. associate-/l* 25.6%

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

   7. Applied egg-rr 25.6%

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

      1. unpow2 25.6%

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

      2. rem-sqrt-square 41.9%

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

   9. Simplified 41.9%

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

   if -0.0200000000000000004 < (sin.f64 kx) < 2.00000000000000003e-119

   1. Initial program 88.2%

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

      1. unpow2 88.2%

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

      2. sqr-neg 88.2%

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

      3. sin-neg 88.2%

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

      4. sin-neg 88.2%

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

      5. unpow2 88.2%

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

      6. associate-*l/ 85.3%

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

      7. associate-/l* 88.1%

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

      8. +-commutative 88.1%

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

      9. unpow2 88.1%

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

      10. sin-neg 88.1%

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

      11. sin-neg 88.1%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 40.6%

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

   if 2.00000000000000003e-119 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 97.7%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 97.7%

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

      2. hypot-undefine 97.7%

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

      3. unpow2 97.7%

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

      4. unpow2 97.7%

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

      5. +-commutative 97.7%

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

      6. associate-*l/ 99.5%

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

      7. *-commutative 99.5%

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

      8. clear-num 99.4%

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

      9. un-div-inv 99.5%

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

      10. +-commutative 99.5%

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

      11. unpow2 99.5%

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

      12. unpow2 99.5%

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

      13. hypot-undefine 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. expm1-log1p-u 99.5%

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

      2. expm1-undefine 90.0%

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

   8. Applied egg-rr 90.0%

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

      1. expm1-define 99.5%

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

   10. Simplified 99.5%

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

   11. Taylor expanded in ky around 0 54.1%

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

Alternative 6: 44.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin th}{\sin kx}\\ \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;\left|ky \cdot t\_1\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-119}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin th) (sin kx))))
   (if (<= (sin kx) -0.02)
     (fabs (* ky t_1))
     (if (<= (sin kx) 2e-119) (sin th) (* (sin ky) t_1)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) / sin(kx);
	double tmp;
	if (sin(kx) <= -0.02) {
		tmp = fabs((ky * t_1));
	} else if (sin(kx) <= 2e-119) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(th) / sin(kx)
    if (sin(kx) <= (-0.02d0)) then
        tmp = abs((ky * t_1))
    else if (sin(kx) <= 2d-119) then
        tmp = sin(th)
    else
        tmp = sin(ky) * t_1
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) / Math.sin(kx);
	double tmp;
	if (Math.sin(kx) <= -0.02) {
		tmp = Math.abs((ky * t_1));
	} else if (Math.sin(kx) <= 2e-119) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) / math.sin(kx)
	tmp = 0
	if math.sin(kx) <= -0.02:
		tmp = math.fabs((ky * t_1))
	elif math.sin(kx) <= 2e-119:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) / sin(kx))
	tmp = 0.0
	if (sin(kx) <= -0.02)
		tmp = abs(Float64(ky * t_1));
	elseif (sin(kx) <= 2e-119)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) / sin(kx);
	tmp = 0.0;
	if (sin(kx) <= -0.02)
		tmp = abs((ky * t_1));
	elseif (sin(kx) <= 2e-119)
		tmp = sin(th);
	else
		tmp = sin(ky) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], N[Abs[N[(ky * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-119], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0200000000000000004

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 14.7%

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

      1. add-sqr-sqrt 12.0%

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

      2. sqrt-unprod 25.6%

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

      3. pow2 25.6%

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

      4. associate-/l* 25.6%

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

   7. Applied egg-rr 25.6%

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

      1. unpow2 25.6%

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

      2. rem-sqrt-square 41.9%

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

   9. Simplified 41.9%

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

   if -0.0200000000000000004 < (sin.f64 kx) < 2.00000000000000003e-119

   1. Initial program 88.2%

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

      1. unpow2 88.2%

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

      2. sqr-neg 88.2%

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

      3. sin-neg 88.2%

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

      4. sin-neg 88.2%

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

      5. unpow2 88.2%

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

      6. associate-*l/ 85.3%

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

      7. associate-/l* 88.1%

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

      8. +-commutative 88.1%

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

      9. unpow2 88.1%

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

      10. sin-neg 88.1%

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

      11. sin-neg 88.1%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 40.6%

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

   if 2.00000000000000003e-119 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 97.7%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 54.1%

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

Alternative 7: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

   1. unpow2 95.1%

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

   2. sqr-neg 95.1%

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

   3. sin-neg 95.1%

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

   4. sin-neg 95.1%

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

   5. unpow2 95.1%

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

   6. associate-*l/ 93.3%

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

   7. associate-/l* 95.1%

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

   8. +-commutative 95.1%

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

   9. unpow2 95.1%

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

   10. sin-neg 95.1%

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

   11. sin-neg 95.1%

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

3. Simplified 99.6%

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

Alternative 8: 64.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.005:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;ky \leq 5.5 \cdot 10^{+204}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.005)
   (* (sin ky) (/ (sin th) (hypot ky (sin kx))))
   (if (<= ky 5.5e+204)
     (* (sin ky) (/ (sin th) (fabs (sin ky))))
     (/ th (/ (hypot (sin ky) (sin kx)) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.005) {
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	} else if (ky <= 5.5e+204) {
		tmp = sin(ky) * (sin(th) / fabs(sin(ky)));
	} else {
		tmp = th / (hypot(sin(ky), sin(kx)) / sin(ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.005) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else if (ky <= 5.5e+204) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.abs(Math.sin(ky)));
	} else {
		tmp = th / (Math.hypot(Math.sin(ky), Math.sin(kx)) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.005:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	elif ky <= 5.5e+204:
		tmp = math.sin(ky) * (math.sin(th) / math.fabs(math.sin(ky)))
	else:
		tmp = th / (math.hypot(math.sin(ky), math.sin(kx)) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.005)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(ky, sin(kx))));
	elseif (ky <= 5.5e+204)
		tmp = Float64(sin(ky) * Float64(sin(th) / abs(sin(ky))));
	else
		tmp = Float64(th / Float64(hypot(sin(ky), sin(kx)) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.005)
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	elseif (ky <= 5.5e+204)
		tmp = sin(ky) * (sin(th) / abs(sin(ky)));
	else
		tmp = th / (hypot(sin(ky), sin(kx)) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.005], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 5.5e+204], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(th / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 0.0050000000000000001

   1. Initial program 93.8%

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

      1. unpow2 93.8%

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

      2. sqr-neg 93.8%

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

      3. sin-neg 93.8%

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

      4. sin-neg 93.8%

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

      5. unpow2 93.8%

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

      6. associate-*l/ 91.5%

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

      7. associate-/l* 93.8%

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

      8. +-commutative 93.8%

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

      9. unpow2 93.8%

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

      10. sin-neg 93.8%

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

      11. sin-neg 93.8%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 64.1%

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

   if 0.0050000000000000001 < ky < 5.4999999999999996e204

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 28.5%

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

      1. add-sqr-sqrt 27.3%

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

      2. sqrt-prod 58.0%

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

      3. rem-sqrt-square 58.0%

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

   7. Applied egg-rr 58.0%

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

   if 5.4999999999999996e204 < ky

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

      2. hypot-undefine 99.6%

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

      3. unpow2 99.6%

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

      4. unpow2 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-*l/ 99.7%

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

      7. *-commutative 99.7%

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

      8. clear-num 99.8%

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

      9. un-div-inv 99.9%

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

      10. +-commutative 99.9%

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

      11. unpow2 99.9%

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

      12. unpow2 99.9%

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

      13. hypot-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in th around 0 72.0%

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

Alternative 9: 64.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.00185:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;ky \leq 2.7 \cdot 10^{+205}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.00185)
   (* (sin ky) (/ (sin th) (hypot ky (sin kx))))
   (if (<= ky 2.7e+205)
     (* (sin ky) (/ (sin th) (fabs (sin ky))))
     (* (sin ky) (/ th (hypot (sin ky) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.00185) {
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	} else if (ky <= 2.7e+205) {
		tmp = sin(ky) * (sin(th) / fabs(sin(ky)));
	} else {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.00185) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else if (ky <= 2.7e+205) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.abs(Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.00185:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	elif ky <= 2.7e+205:
		tmp = math.sin(ky) * (math.sin(th) / math.fabs(math.sin(ky)))
	else:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.00185)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(ky, sin(kx))));
	elseif (ky <= 2.7e+205)
		tmp = Float64(sin(ky) * Float64(sin(th) / abs(sin(ky))));
	else
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.00185)
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	elseif (ky <= 2.7e+205)
		tmp = sin(ky) * (sin(th) / abs(sin(ky)));
	else
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.00185], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 2.7e+205], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 0.0018500000000000001

   1. Initial program 93.8%

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

      1. unpow2 93.8%

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

      2. sqr-neg 93.8%

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

      3. sin-neg 93.8%

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

      4. sin-neg 93.8%

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

      5. unpow2 93.8%

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

      6. associate-*l/ 91.5%

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

      7. associate-/l* 93.8%

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

      8. +-commutative 93.8%

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

      9. unpow2 93.8%

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

      10. sin-neg 93.8%

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

      11. sin-neg 93.8%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 64.1%

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

   if 0.0018500000000000001 < ky < 2.70000000000000012e205

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 28.5%

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

      1. add-sqr-sqrt 27.3%

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

      2. sqrt-prod 58.0%

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

      3. rem-sqrt-square 58.0%

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

   7. Applied egg-rr 58.0%

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

   if 2.70000000000000012e205 < ky

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0 71.6%

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

Alternative 10: 42.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-119}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.02)
   (fabs (* ky (/ (sin th) (sin kx))))
   (if (<= (sin kx) 4e-119) (sin th) (/ (sin th) (/ (sin kx) ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.02) {
		tmp = fabs((ky * (sin(th) / sin(kx))));
	} else if (sin(kx) <= 4e-119) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(kx) / ky);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.02d0)) then
        tmp = abs((ky * (sin(th) / sin(kx))))
    else if (sin(kx) <= 4d-119) then
        tmp = sin(th)
    else
        tmp = sin(th) / (sin(kx) / ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.02) {
		tmp = Math.abs((ky * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(kx) <= 4e-119) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.02:
		tmp = math.fabs((ky * (math.sin(th) / math.sin(kx))))
	elif math.sin(kx) <= 4e-119:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.02)
		tmp = abs(Float64(ky * Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= 4e-119)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.02)
		tmp = abs((ky * (sin(th) / sin(kx))));
	elseif (sin(kx) <= 4e-119)
		tmp = sin(th);
	else
		tmp = sin(th) / (sin(kx) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], N[Abs[N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-119], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0200000000000000004

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 14.7%

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

      1. add-sqr-sqrt 12.0%

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

      2. sqrt-unprod 25.6%

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

      3. pow2 25.6%

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

      4. associate-/l* 25.6%

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

   7. Applied egg-rr 25.6%

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

      1. unpow2 25.6%

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

      2. rem-sqrt-square 41.9%

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

   9. Simplified 41.9%

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

   if -0.0200000000000000004 < (sin.f64 kx) < 4.00000000000000005e-119

   1. Initial program 88.2%

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

      1. unpow2 88.2%

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

      2. sqr-neg 88.2%

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

      3. sin-neg 88.2%

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

      4. sin-neg 88.2%

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

      5. unpow2 88.2%

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

      6. associate-*l/ 85.3%

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

      7. associate-/l* 88.1%

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

      8. +-commutative 88.1%

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

      9. unpow2 88.1%

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

      10. sin-neg 88.1%

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

      11. sin-neg 88.1%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 40.6%

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

   if 4.00000000000000005e-119 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 97.7%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 97.7%

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

      2. hypot-undefine 97.7%

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

      3. unpow2 97.7%

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

      4. unpow2 97.7%

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

      5. +-commutative 97.7%

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

      6. associate-*l/ 99.5%

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

      7. *-commutative 99.5%

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

      8. clear-num 99.4%

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

      9. un-div-inv 99.5%

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

      10. +-commutative 99.5%

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

      11. unpow2 99.5%

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

      12. unpow2 99.5%

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

      13. hypot-undefine 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ky around 0 45.6%

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

Alternative 11: 46.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.0005:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 10^{-35}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.0005)
   (sqrt (pow (sin th) 2.0))
   (if (<= (sin ky) 1e-35) (* ky (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.0005) {
		tmp = sqrt(pow(sin(th), 2.0));
	} else if (sin(ky) <= 1e-35) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.0005d0)) then
        tmp = sqrt((sin(th) ** 2.0d0))
    else if (sin(ky) <= 1d-35) then
        tmp = ky * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.0005) {
		tmp = Math.sqrt(Math.pow(Math.sin(th), 2.0));
	} else if (Math.sin(ky) <= 1e-35) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.0005:
		tmp = math.sqrt(math.pow(math.sin(th), 2.0))
	elif math.sin(ky) <= 1e-35:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.0005)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 1e-35)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.0005)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 1e-35)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.0005], N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-35], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -5.0000000000000001e-4

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 2.9%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 26.5%

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

      3. pow2 26.5%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   7. Applied egg-rr 26.5%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]

   if -5.0000000000000001e-4 < (sin.f64 ky) < 1.00000000000000001e-35

   1. Initial program 89.6%

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

      1. unpow2 89.6%

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

      2. sqr-neg 89.6%

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

      3. sin-neg 89.6%

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

      4. sin-neg 89.6%

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

      5. unpow2 89.6%

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

      6. associate-*l/ 85.8%

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

      7. associate-/l* 89.7%

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

      8. +-commutative 89.7%

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

      9. unpow2 89.7%

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

      10. sin-neg 89.7%

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

      11. sin-neg 89.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 47.2%

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

      1. associate-/l* 49.6%

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

   7. Simplified 49.6%

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

   if 1.00000000000000001e-35 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 61.2%

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

Alternative 12: 47.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.0005:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-35}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.0005)
   (fabs (sin th))
   (if (<= (sin ky) 1e-35) (* ky (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.0005) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-35) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.0005d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-35) then
        tmp = ky * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.0005) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-35) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.0005:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-35:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.0005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-35)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.0005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-35)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.0005], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-35], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -5.0000000000000001e-4

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 2.9%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 26.5%

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

      3. pow2 26.5%

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

      4. clear-num 26.5%

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

      5. un-div-inv 26.5%

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

   7. Applied egg-rr 26.5%

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

      1. unpow2 26.5%

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

      2. rem-sqrt-square 34.5%

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

      3. associate-/r/ 34.6%

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

      4. *-inverses 34.6%

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

      5. *-lft-identity 34.6%

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

   9. Simplified 34.6%

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

   if -5.0000000000000001e-4 < (sin.f64 ky) < 1.00000000000000001e-35

   1. Initial program 89.6%

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

      1. unpow2 89.6%

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

      2. sqr-neg 89.6%

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

      3. sin-neg 89.6%

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

      4. sin-neg 89.6%

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

      5. unpow2 89.6%

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

      6. associate-*l/ 85.8%

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

      7. associate-/l* 89.7%

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

      8. +-commutative 89.7%

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

      9. unpow2 89.7%

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

      10. sin-neg 89.7%

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

      11. sin-neg 89.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 47.2%

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

      1. associate-/l* 49.6%

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

   7. Simplified 49.6%

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

   if 1.00000000000000001e-35 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 61.2%

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

Alternative 13: 61.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.37:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\left|\sin kx\right|}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.37)
   (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
   (/ (sin ky) (/ (fabs (sin kx)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.37) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(ky) / (fabs(sin(kx)) / sin(th));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.37) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.sin(ky) / (Math.abs(Math.sin(kx)) / Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.37:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.sin(ky) / (math.fabs(math.sin(kx)) / math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.37)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(sin(ky) / Float64(abs(sin(kx)) / sin(th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.37)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(ky) / (abs(sin(kx)) / sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.37], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.37

   1. Initial program 96.2%

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

      1. unpow2 96.2%

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

      2. sqr-neg 96.2%

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

      3. sin-neg 96.2%

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

      4. sin-neg 96.2%

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

      5. unpow2 96.2%

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

      6. associate-*l/ 93.8%

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

      7. associate-/l* 96.2%

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

      8. +-commutative 96.2%

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

      9. unpow2 96.2%

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

      10. sin-neg 96.2%

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

      11. sin-neg 96.2%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0 68.0%

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

   if 0.37 < th

   1. Initial program 92.1%

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

      1. unpow2 92.1%

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

      2. sqr-neg 92.1%

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

      3. sin-neg 92.1%

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

      4. sin-neg 92.1%

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

      5. unpow2 92.1%

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

      6. associate-*l/ 92.2%

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

      7. associate-/l* 92.2%

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

      8. +-commutative 92.2%

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

      9. unpow2 92.2%

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

      10. sin-neg 92.2%

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

      11. sin-neg 92.2%

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

   3. Simplified 99.7%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ky around 0 23.7%

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

      1. add-sqr-sqrt 22.5%

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

      2. sqrt-prod 43.7%

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

      3. rem-sqrt-square 51.3%

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

   9. Applied egg-rr 51.3%

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

Alternative 14: 55.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0018:\\ \;\;\;\;\frac{\sin ky}{\frac{\left|\sin kx\right|}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0018)
   (/ (sin ky) (/ (fabs (sin kx)) (sin th)))
   (* (sin ky) (/ (sin th) (fabs (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0018) {
		tmp = sin(ky) / (fabs(sin(kx)) / sin(th));
	} else {
		tmp = sin(ky) * (sin(th) / fabs(sin(ky)));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 0.0018d0) then
        tmp = sin(ky) / (abs(sin(kx)) / sin(th))
    else
        tmp = sin(ky) * (sin(th) / abs(sin(ky)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0018) {
		tmp = Math.sin(ky) / (Math.abs(Math.sin(kx)) / Math.sin(th));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.abs(Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.0018:
		tmp = math.sin(ky) / (math.fabs(math.sin(kx)) / math.sin(th))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.fabs(math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0018)
		tmp = Float64(sin(ky) / Float64(abs(sin(kx)) / sin(th)));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / abs(sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.0018)
		tmp = sin(ky) / (abs(sin(kx)) / sin(th));
	else
		tmp = sin(ky) * (sin(th) / abs(sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0018], N[(N[Sin[ky], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.0018

   1. Initial program 93.8%

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

      1. unpow2 93.8%

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

      2. sqr-neg 93.8%

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

      3. sin-neg 93.8%

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

      4. sin-neg 93.8%

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

      5. unpow2 93.8%

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

      6. associate-*l/ 91.5%

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

      7. associate-/l* 93.8%

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

      8. +-commutative 93.8%

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

      9. unpow2 93.8%

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

      10. sin-neg 93.8%

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

      11. sin-neg 93.8%

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

   3. Simplified 99.6%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in ky around 0 33.6%

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

      1. add-sqr-sqrt 27.6%

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

      2. sqrt-prod 51.9%

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

      3. rem-sqrt-square 56.1%

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

   9. Applied egg-rr 56.1%

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

   if 0.0018 < ky

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 25.1%

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

      1. add-sqr-sqrt 23.6%

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

      2. sqrt-prod 56.6%

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

      3. rem-sqrt-square 56.6%

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

   7. Applied egg-rr 56.6%

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

Alternative 15: 50.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0018:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0018)
   (/ (* (sin th) ky) (fabs (sin kx)))
   (* (sin ky) (/ (sin th) (fabs (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0018) {
		tmp = (sin(th) * ky) / fabs(sin(kx));
	} else {
		tmp = sin(ky) * (sin(th) / fabs(sin(ky)));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 0.0018d0) then
        tmp = (sin(th) * ky) / abs(sin(kx))
    else
        tmp = sin(ky) * (sin(th) / abs(sin(ky)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0018) {
		tmp = (Math.sin(th) * ky) / Math.abs(Math.sin(kx));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.abs(Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.0018:
		tmp = (math.sin(th) * ky) / math.fabs(math.sin(kx))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.fabs(math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0018)
		tmp = Float64(Float64(sin(th) * ky) / abs(sin(kx)));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / abs(sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.0018)
		tmp = (sin(th) * ky) / abs(sin(kx));
	else
		tmp = sin(ky) * (sin(th) / abs(sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0018], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.0018

   1. Initial program 93.8%

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

      1. unpow2 93.8%

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

      2. sqr-neg 93.8%

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

      3. sin-neg 93.8%

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

      4. sin-neg 93.8%

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

      5. unpow2 93.8%

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

      6. associate-*l/ 91.5%

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

      7. associate-/l* 93.8%

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

      8. +-commutative 93.8%

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

      9. unpow2 93.8%

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

      10. sin-neg 93.8%

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

      11. sin-neg 93.8%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 29.6%

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

      1. add-sqr-sqrt 27.6%

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

      2. sqrt-prod 51.9%

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

      3. rem-sqrt-square 56.1%

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

   7. Applied egg-rr 50.3%

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

   if 0.0018 < ky

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 25.1%

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

      1. add-sqr-sqrt 23.6%

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

      2. sqrt-prod 56.6%

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

      3. rem-sqrt-square 56.6%

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

   7. Applied egg-rr 56.6%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0018:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.1:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-68}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.1)
   (fabs (sin th))
   (if (<= (sin ky) 1e-68) (* ky (/ (sin th) kx)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.1) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-68) {
		tmp = ky * (sin(th) / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.1d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-68) then
        tmp = ky * (sin(th) / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.1) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-68) {
		tmp = ky * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.1:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-68:
		tmp = ky * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.1)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-68)
		tmp = Float64(ky * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.1)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-68)
		tmp = ky * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.1], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-68], N[(ky * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.10000000000000001

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 2.8%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 29.1%

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

      3. pow2 29.1%

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

      4. clear-num 29.1%

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

      5. un-div-inv 29.2%

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

   7. Applied egg-rr 29.2%

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

      1. unpow2 29.2%

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

      2. rem-sqrt-square 38.3%

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

      3. associate-/r/ 38.3%

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

      4. *-inverses 38.3%

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

      5. *-lft-identity 38.3%

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

   9. Simplified 38.3%

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

   if -0.10000000000000001 < (sin.f64 ky) < 1.00000000000000007e-68

   1. Initial program 89.5%

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

      1. unpow2 89.5%

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

      2. sqr-neg 89.5%

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

      3. sin-neg 89.5%

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

      4. sin-neg 89.5%

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

      5. unpow2 89.5%

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

      6. associate-*l/ 85.7%

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

      7. associate-/l* 89.6%

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

      8. +-commutative 89.6%

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

      9. unpow2 89.6%

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

      10. sin-neg 89.6%

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

      11. sin-neg 89.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 47.4%

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

   6. Taylor expanded in kx around 0 28.0%

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

      1. associate-/l* 30.4%

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

   8. Simplified 30.4%

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

   if 1.00000000000000007e-68 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. +-commutative 99.7%

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

      9. unpow2 99.7%

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

      10. sin-neg 99.7%

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

      11. sin-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 58.6%

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

Alternative 17: 25.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 6.5 \cdot 10^{-69}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 6.5e-69) (* ky (/ (sin th) kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 6.5e-69) {
		tmp = ky * (sin(th) / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 6.5d-69) then
        tmp = ky * (sin(th) / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 6.5e-69) {
		tmp = ky * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 6.5e-69:
		tmp = ky * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 6.5e-69)
		tmp = Float64(ky * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 6.5e-69)
		tmp = ky * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 6.5e-69], N[(ky * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 6.49999999999999951e-69

   1. Initial program 93.3%

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

      1. unpow2 93.3%

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

      2. sqr-neg 93.3%

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

      3. sin-neg 93.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 93.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 93.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 90.9%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 93.3%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. +-commutative 93.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin \left(-kx\right)}^{2}}}} \]

      9. unpow2 93.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)}}} \]

      10. sin-neg 93.3%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right)}} \]

      11. sin-neg 93.3%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 30.7%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]

   6. Taylor expanded in kx around 0 19.0%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]
   7. Step-by-step derivation

      1. associate-/l* 20.5%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{kx}} \]

   8. Simplified 20.5%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{kx}} \]

   if 6.49999999999999951e-69 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 99.7%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 99.7%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 99.7%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 99.7%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 99.7%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 99.6%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. +-commutative 99.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin \left(-kx\right)}^{2}}}} \]

      9. unpow2 99.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)}}} \]

      10. sin-neg 99.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right)}} \]

      11. sin-neg 99.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 30.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 27.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 6.2 \cdot 10^{-89}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 6.2e-89) (sin th) (* ky (/ th (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.2e-89) {
		tmp = sin(th);
	} else {
		tmp = ky * (th / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 6.2d-89) then
        tmp = sin(th)
    else
        tmp = ky * (th / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.2e-89) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (th / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 6.2e-89:
		tmp = math.sin(th)
	else:
		tmp = ky * (th / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 6.2e-89)
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(th / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 6.2e-89)
		tmp = sin(th);
	else
		tmp = ky * (th / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 6.2e-89], N[Sin[th], $MachinePrecision], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 6.19999999999999993e-89

   1. Initial program 93.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 93.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 93.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 93.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 93.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 93.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 91.4%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 93.4%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. +-commutative 93.4%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin \left(-kx\right)}^{2}}}} \]

      9. unpow2 93.4%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)}}} \]

      10. sin-neg 93.4%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right)}} \]

      11. sin-neg 93.4%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 25.5%

      \[\leadsto \color{blue}{\sin th} \]

   if 6.19999999999999993e-89 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 98.3%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 99.5%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. +-commutative 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin \left(-kx\right)}^{2}}}} \]

      9. unpow2 99.5%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)}}} \]

      10. sin-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right)}} \]

      11. sin-neg 99.5%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)}}} \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in th around 0 58.7%

      \[\leadsto \sin ky \cdot \color{blue}{\left(th \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. rem-square-sqrt 58.8%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}}\right) \]

      2. +-commutative 58.8%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \]

      3. unpow2 58.8%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \]

      4. unpow2 58.8%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \]

      5. hypot-undefine 58.8%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \]

      6. +-commutative 58.8%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}}\right) \]

      7. unpow2 58.8%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}}\right) \]

      8. unpow2 58.8%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}}\right) \]

      9. hypot-undefine 58.8%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right) \]

      10. unpow2 58.8%

          \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\color{blue}{{\left(\mathsf{hypot}\left(\sin ky, \sin kx\right)\right)}^{2}}}}\right) \]

   7. Simplified 58.8%

      \[\leadsto \sin ky \cdot \color{blue}{\left(th \cdot \sqrt{\frac{1}{{\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}}}\right)} \]

   8. Taylor expanded in ky around 0 18.5%

      \[\leadsto \color{blue}{\frac{ky \cdot th}{\sin kx}} \]
   9. Step-by-step derivation

      1. associate-/l* 19.7%

         \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   10. Simplified 19.7%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 23.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.9 \cdot 10^{-126}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.9e-126) (* ky (/ th kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.9e-126) {
		tmp = ky * (th / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 1.9d-126) then
        tmp = ky * (th / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.9e-126) {
		tmp = ky * (th / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.9e-126:
		tmp = ky * (th / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.9e-126)
		tmp = Float64(ky * Float64(th / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.9e-126)
		tmp = ky * (th / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.9e-126], N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.8999999999999999e-126

   1. Initial program 92.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 92.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 92.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 92.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 92.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 92.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 90.3%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 92.9%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. +-commutative 92.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin \left(-kx\right)}^{2}}}} \]

      9. unpow2 92.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)}}} \]

      10. sin-neg 92.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right)}} \]

      11. sin-neg 92.9%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in th around 0 49.6%

      \[\leadsto \sin ky \cdot \color{blue}{\left(th \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. rem-square-sqrt 49.6%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}}\right) \]

      2. +-commutative 49.6%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \]

      3. unpow2 49.6%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \]

      4. unpow2 49.6%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \]

      5. hypot-undefine 49.6%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \]

      6. +-commutative 49.6%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}}\right) \]

      7. unpow2 49.6%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}}\right) \]

      8. unpow2 49.6%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}}\right) \]

      9. hypot-undefine 49.6%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right) \]

      10. unpow2 49.6%

          \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\color{blue}{{\left(\mathsf{hypot}\left(\sin ky, \sin kx\right)\right)}^{2}}}}\right) \]

   7. Simplified 49.6%

      \[\leadsto \sin ky \cdot \color{blue}{\left(th \cdot \sqrt{\frac{1}{{\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}}}\right)} \]

   8. Taylor expanded in ky around 0 15.7%

      \[\leadsto \color{blue}{\frac{ky \cdot th}{\sin kx}} \]
   9. Step-by-step derivation

      1. associate-/l* 17.3%

         \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   10. Simplified 17.3%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   11. Taylor expanded in kx around 0 14.5%

       \[\leadsto \color{blue}{\frac{ky \cdot th}{kx}} \]
   12. Step-by-step derivation

       1. associate-/l* 16.1%

          \[\leadsto \color{blue}{ky \cdot \frac{th}{kx}} \]

   13. Simplified 16.1%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{kx}} \]

   if 1.8999999999999999e-126 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 99.7%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. +-commutative 99.7%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin \left(-kx\right)}^{2}}}} \]

      9. unpow2 99.7%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)}}} \]

      10. sin-neg 99.7%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right)}} \]

      11. sin-neg 99.7%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 28.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 17.6% accurate, 70.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.95 \cdot 10^{-32}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.95e-32) (* ky (/ th kx)) th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.95e-32) {
		tmp = ky * (th / 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 <= 1.95d-32) then
        tmp = ky * (th / 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 <= 1.95e-32) {
		tmp = ky * (th / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.95e-32:
		tmp = ky * (th / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.95e-32)
		tmp = Float64(ky * Float64(th / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.95e-32)
		tmp = ky * (th / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.95e-32], N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if ky < 1.9500000000000001e-32

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 93.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 93.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 93.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 93.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 93.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 91.3%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 93.7%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. +-commutative 93.7%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin \left(-kx\right)}^{2}}}} \]

      9. unpow2 93.7%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)}}} \]

      10. sin-neg 93.7%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right)}} \]

      11. sin-neg 93.7%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in th around 0 49.1%

      \[\leadsto \sin ky \cdot \color{blue}{\left(th \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. rem-square-sqrt 49.0%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}}\right) \]

      2. +-commutative 49.0%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \]

      3. unpow2 49.0%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \]

      4. unpow2 49.0%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \]

      5. hypot-undefine 49.0%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \]

      6. +-commutative 49.0%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}}\right) \]

      7. unpow2 49.0%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}}\right) \]

      8. unpow2 49.0%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}}\right) \]

      9. hypot-undefine 49.0%

         \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}\right) \]

      10. unpow2 49.0%

          \[\leadsto \sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\color{blue}{{\left(\mathsf{hypot}\left(\sin ky, \sin kx\right)\right)}^{2}}}}\right) \]

   7. Simplified 49.0%

      \[\leadsto \sin ky \cdot \color{blue}{\left(th \cdot \sqrt{\frac{1}{{\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}}}\right)} \]

   8. Taylor expanded in ky around 0 15.5%

      \[\leadsto \color{blue}{\frac{ky \cdot th}{\sin kx}} \]
   9. Step-by-step derivation

      1. associate-/l* 16.9%

         \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   10. Simplified 16.9%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   11. Taylor expanded in kx around 0 14.5%

       \[\leadsto \color{blue}{\frac{ky \cdot th}{kx}} \]
   12. Step-by-step derivation

       1. associate-/l* 15.9%

          \[\leadsto \color{blue}{ky \cdot \frac{th}{kx}} \]

   13. Simplified 15.9%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{kx}} \]

   if 1.9500000000000001e-32 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 99.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 99.6%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. +-commutative 99.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin \left(-kx\right)}^{2}}}} \]

      9. unpow2 99.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)}}} \]

      10. sin-neg 99.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right)}} \]

      11. sin-neg 99.6%

          \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 28.3%

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]

   6. Taylor expanded in th around 0 11.2%

      \[\leadsto \color{blue}{th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 14.0% 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 95.1%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. unpow2 95.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

   2. sqr-neg 95.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   3. sin-neg 95.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. sin-neg 95.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. unpow2 95.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

   6. associate-*l/ 93.3%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

   7. associate-/l* 95.1%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

   8. +-commutative 95.1%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin \left(-kx\right)}^{2}}}} \]

   9. unpow2 95.1%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)}}} \]

   10. sin-neg 95.1%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right)} \cdot \sin \left(-kx\right)}} \]

   11. sin-neg 95.1%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin ky}^{2} + \left(-\sin kx\right) \cdot \color{blue}{\left(-\sin kx\right)}}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
4. Add Preprocessing

5. Taylor expanded in kx around 0 21.4%

   \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]

6. Taylor expanded in th around 0 11.2%

   \[\leadsto \color{blue}{th} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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)))