Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.6%

Time: 16.3s

Alternatives: 19

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.4× speedup

Math

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

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 93.2%

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

   1. unpow2 93.2%

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

   2. sqr-neg 93.2%

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

   3. sin-neg 93.2%

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

   4. sin-neg 93.2%

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

   5. unpow2 93.2%

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

   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* 93.2%

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

   8. unpow2 93.2%

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

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/ 95.3%

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

   2. hypot-undefine 91.3%

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

   3. unpow2 91.3%

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

   4. unpow2 91.3%

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

   5. +-commutative 91.3%

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

   6. associate-*l/ 93.2%

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

   7. *-commutative 93.2%

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

   8. clear-num 93.1%

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

   9. un-div-inv 93.2%

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

   10. +-commutative 93.2%

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

   11. unpow2 93.2%

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

   12. unpow2 93.2%

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

   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. Final simplification 99.7%

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

Alternative 2: 46.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.025:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-118}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-16}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.025)
   (/ (* th (sin ky)) (hypot (sin ky) kx))
   (if (<= (sin ky) 5e-118)
     (* (sin ky) (/ (sin th) (sin kx)))
     (if (<= (sin ky) 1e-16)
       (/ (* th ky) (hypot (sin ky) (sin kx)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.025) {
		tmp = (th * sin(ky)) / hypot(sin(ky), kx);
	} else if (sin(ky) <= 5e-118) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (sin(ky) <= 1e-16) {
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.025) {
		tmp = (th * Math.sin(ky)) / Math.hypot(Math.sin(ky), kx);
	} else if (Math.sin(ky) <= 5e-118) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(ky) <= 1e-16) {
		tmp = (th * ky) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.025:
		tmp = (th * math.sin(ky)) / math.hypot(math.sin(ky), kx)
	elif math.sin(ky) <= 5e-118:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif math.sin(ky) <= 1e-16:
		tmp = (th * ky) / math.hypot(math.sin(ky), math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.025)
		tmp = Float64(Float64(th * sin(ky)) / hypot(sin(ky), kx));
	elseif (sin(ky) <= 5e-118)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (sin(ky) <= 1e-16)
		tmp = Float64(Float64(th * ky) / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.025)
		tmp = (th * sin(ky)) / hypot(sin(ky), kx);
	elseif (sin(ky) <= 5e-118)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (sin(ky) <= 1e-16)
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.025], 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], 5e-118], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-16], N[(N[(th * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.025000000000000001

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.7%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   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/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in th around 0 58.4%

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

   8. Taylor expanded in kx around 0 32.3%

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

   if -0.025000000000000001 < (sin.f64 ky) < 5.00000000000000015e-118

   1. Initial program 83.7%

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

      1. unpow2 83.7%

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

      2. sqr-neg 83.7%

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

      3. sin-neg 83.7%

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

      4. sin-neg 83.7%

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

      5. unpow2 83.7%

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

      6. associate-*l/ 80.1%

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

      7. associate-/l* 83.7%

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

      8. unpow2 83.7%

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

   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 46.3%

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

   if 5.00000000000000015e-118 < (sin.f64 ky) < 9.9999999999999998e-17

   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/ 95.1%

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

      7. associate-/l* 99.8%

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

      8. unpow2 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*r/ 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in th around 0 80.5%

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

   8. Taylor expanded in ky around 0 80.5%

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

   if 9.9999999999999998e-17 < (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/ 99.7%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   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 55.5%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.025:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-118}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-16}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.025:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\left|th \cdot \sin ky\right|}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-118}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-16}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.025)
   (/ 1.0 (/ (sin ky) (fabs (* th (sin ky)))))
   (if (<= (sin ky) 5e-118)
     (* (sin ky) (/ (sin th) (sin kx)))
     (if (<= (sin ky) 1e-16)
       (/ (* th ky) (hypot (sin ky) (sin kx)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.025) {
		tmp = 1.0 / (sin(ky) / fabs((th * sin(ky))));
	} else if (sin(ky) <= 5e-118) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (sin(ky) <= 1e-16) {
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.025) {
		tmp = 1.0 / (Math.sin(ky) / Math.abs((th * Math.sin(ky))));
	} else if (Math.sin(ky) <= 5e-118) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(ky) <= 1e-16) {
		tmp = (th * ky) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.025:
		tmp = 1.0 / (math.sin(ky) / math.fabs((th * math.sin(ky))))
	elif math.sin(ky) <= 5e-118:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif math.sin(ky) <= 1e-16:
		tmp = (th * ky) / math.hypot(math.sin(ky), math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.025)
		tmp = Float64(1.0 / Float64(sin(ky) / abs(Float64(th * sin(ky)))));
	elseif (sin(ky) <= 5e-118)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (sin(ky) <= 1e-16)
		tmp = Float64(Float64(th * ky) / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.025)
		tmp = 1.0 / (sin(ky) / abs((th * sin(ky))));
	elseif (sin(ky) <= 5e-118)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (sin(ky) <= 1e-16)
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.025], N[(1.0 / N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-118], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-16], N[(N[(th * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.025000000000000001

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.7%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.7%

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

      1. associate-*r/ 2.7%

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

      2. clear-num 2.7%

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

      3. *-commutative 2.7%

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

   7. Applied egg-rr 2.7%

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

   8. Taylor expanded in th around 0 3.2%

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

      1. add-sqr-sqrt 1.6%

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

      2. sqrt-unprod 10.3%

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

      3. pow2 10.3%

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

      4. *-commutative 10.3%

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

   10. Applied egg-rr 10.3%

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

       1. unpow2 10.3%

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

       2. rem-sqrt-square 19.0%

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

       3. *-commutative 19.0%

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

   12. Simplified 19.0%

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

   if -0.025000000000000001 < (sin.f64 ky) < 5.00000000000000015e-118

   1. Initial program 83.7%

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

      1. unpow2 83.7%

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

      2. sqr-neg 83.7%

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

      3. sin-neg 83.7%

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

      4. sin-neg 83.7%

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

      5. unpow2 83.7%

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

      6. associate-*l/ 80.1%

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

      7. associate-/l* 83.7%

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

      8. unpow2 83.7%

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

   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 46.3%

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

   if 5.00000000000000015e-118 < (sin.f64 ky) < 9.9999999999999998e-17

   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/ 95.1%

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

      7. associate-/l* 99.8%

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

      8. unpow2 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*r/ 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in th around 0 80.5%

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

   8. Taylor expanded in ky around 0 80.5%

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

   if 9.9999999999999998e-17 < (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/ 99.7%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   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 55.5%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.025:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\left|th \cdot \sin ky\right|}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-118}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-16}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.015:\\ \;\;\;\;\left(\sin th \cdot ky\right) \cdot \sqrt{\frac{1}{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-121}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.015)
   (* (* (sin th) ky) (sqrt (/ 1.0 (- 0.5 (* 0.5 (cos (* kx 2.0)))))))
   (if (<= (sin kx) 5e-121)
     (sin th)
     (if (<= (sin kx) 5e-17)
       (/ (* th (sin ky)) (hypot (sin ky) kx))
       (* (sin ky) (/ (sin th) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.015) {
		tmp = (sin(th) * ky) * sqrt((1.0 / (0.5 - (0.5 * cos((kx * 2.0))))));
	} else if (sin(kx) <= 5e-121) {
		tmp = sin(th);
	} else if (sin(kx) <= 5e-17) {
		tmp = (th * sin(ky)) / hypot(sin(ky), kx);
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.015) {
		tmp = (Math.sin(th) * ky) * Math.sqrt((1.0 / (0.5 - (0.5 * Math.cos((kx * 2.0))))));
	} else if (Math.sin(kx) <= 5e-121) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 5e-17) {
		tmp = (th * Math.sin(ky)) / Math.hypot(Math.sin(ky), kx);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.015:
		tmp = (math.sin(th) * ky) * math.sqrt((1.0 / (0.5 - (0.5 * math.cos((kx * 2.0))))))
	elif math.sin(kx) <= 5e-121:
		tmp = math.sin(th)
	elif math.sin(kx) <= 5e-17:
		tmp = (th * math.sin(ky)) / math.hypot(math.sin(ky), kx)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.015)
		tmp = Float64(Float64(sin(th) * ky) * sqrt(Float64(1.0 / Float64(0.5 - Float64(0.5 * cos(Float64(kx * 2.0)))))));
	elseif (sin(kx) <= 5e-121)
		tmp = sin(th);
	elseif (sin(kx) <= 5e-17)
		tmp = Float64(Float64(th * sin(ky)) / hypot(sin(ky), kx));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.015)
		tmp = (sin(th) * ky) * sqrt((1.0 / (0.5 - (0.5 * cos((kx * 2.0))))));
	elseif (sin(kx) <= 5e-121)
		tmp = sin(th);
	elseif (sin(kx) <= 5e-17)
		tmp = (th * sin(ky)) / hypot(sin(ky), kx);
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.015], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(0.5 - N[(0.5 * N[Cos[N[(kx * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-121], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-17], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.014999999999999999

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. sin-mult 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. div-sub 99.0%

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

      2. +-inverses 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      3. cos-0 99.0%

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

      4. metadata-eval 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{0.5} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      5. count-2 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot kx\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      6. *-commutative 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(kx \cdot 2\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

   6. Simplified 99.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \frac{\cos \left(kx \cdot 2\right)}{2}\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0 53.5%

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \]

   if -0.014999999999999999 < (sin.f64 kx) < 4.99999999999999989e-121

   1. Initial program 82.3%

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

      1. unpow2 82.3%

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

      2. sqr-neg 82.3%

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

      3. sin-neg 82.3%

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

      4. sin-neg 82.3%

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

      5. unpow2 82.3%

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

      6. associate-*l/ 78.2%

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

      7. associate-/l* 82.1%

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

      8. unpow2 82.1%

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

   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 31.5%

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

   if 4.99999999999999989e-121 < (sin.f64 kx) < 4.9999999999999999e-17

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 94.8%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 94.8%

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

   6. Applied egg-rr 94.8%

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

   7. Taylor expanded in th around 0 73.6%

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

   8. Taylor expanded in kx around 0 73.6%

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

   if 4.9999999999999999e-17 < (sin.f64 kx)

   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. unpow2 99.6%

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

   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 55.2%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.015:\\ \;\;\;\;\left(\sin th \cdot ky\right) \cdot \sqrt{\frac{1}{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-121}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.015:\\ \;\;\;\;\left(\sin th \cdot ky\right) \cdot \sqrt{\frac{1}{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-59}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.015)
   (* (* (sin th) ky) (sqrt (/ 1.0 (- 0.5 (* 0.5 (cos (* kx 2.0)))))))
   (if (<= (sin kx) 5e-59)
     (* (sin ky) (fabs (/ (sin th) (sin ky))))
     (if (<= (sin kx) 5e-17)
       (/ (* th (sin ky)) (hypot (sin ky) kx))
       (* (sin ky) (/ (sin th) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.015) {
		tmp = (sin(th) * ky) * sqrt((1.0 / (0.5 - (0.5 * cos((kx * 2.0))))));
	} else if (sin(kx) <= 5e-59) {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	} else if (sin(kx) <= 5e-17) {
		tmp = (th * sin(ky)) / hypot(sin(ky), kx);
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.015) {
		tmp = (Math.sin(th) * ky) * Math.sqrt((1.0 / (0.5 - (0.5 * Math.cos((kx * 2.0))))));
	} else if (Math.sin(kx) <= 5e-59) {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	} else if (Math.sin(kx) <= 5e-17) {
		tmp = (th * Math.sin(ky)) / Math.hypot(Math.sin(ky), kx);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.015:
		tmp = (math.sin(th) * ky) * math.sqrt((1.0 / (0.5 - (0.5 * math.cos((kx * 2.0))))))
	elif math.sin(kx) <= 5e-59:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	elif math.sin(kx) <= 5e-17:
		tmp = (th * math.sin(ky)) / math.hypot(math.sin(ky), kx)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.015)
		tmp = Float64(Float64(sin(th) * ky) * sqrt(Float64(1.0 / Float64(0.5 - Float64(0.5 * cos(Float64(kx * 2.0)))))));
	elseif (sin(kx) <= 5e-59)
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	elseif (sin(kx) <= 5e-17)
		tmp = Float64(Float64(th * sin(ky)) / hypot(sin(ky), kx));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.015)
		tmp = (sin(th) * ky) * sqrt((1.0 / (0.5 - (0.5 * cos((kx * 2.0))))));
	elseif (sin(kx) <= 5e-59)
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	elseif (sin(kx) <= 5e-17)
		tmp = (th * sin(ky)) / hypot(sin(ky), kx);
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.015], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(0.5 - N[(0.5 * N[Cos[N[(kx * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-59], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-17], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.014999999999999999

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. sin-mult 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. div-sub 99.0%

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

      2. +-inverses 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      3. cos-0 99.0%

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

      4. metadata-eval 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{0.5} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      5. count-2 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot kx\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      6. *-commutative 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(kx \cdot 2\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

   6. Simplified 99.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \frac{\cos \left(kx \cdot 2\right)}{2}\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0 53.5%

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \]

   if -0.014999999999999999 < (sin.f64 kx) < 5.0000000000000001e-59

   1. Initial program 83.8%

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

      1. unpow2 83.8%

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

      2. sqr-neg 83.8%

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

      3. sin-neg 83.8%

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

      4. sin-neg 83.8%

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

      5. unpow2 83.8%

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

      6. associate-*l/ 79.2%

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

      7. associate-/l* 83.7%

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

      8. unpow2 83.7%

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

   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 32.8%

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

      1. add-sqr-sqrt 16.0%

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

      2. sqrt-unprod 30.0%

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

      3. pow2 30.0%

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

   7. Applied egg-rr 30.0%

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

      1. unpow2 30.0%

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

      2. rem-sqrt-square 41.6%

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

   9. Simplified 41.6%

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

   if 5.0000000000000001e-59 < (sin.f64 kx) < 4.9999999999999999e-17

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 99.7%

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

      7. associate-/l* 99.3%

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

      8. unpow2 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in th around 0 88.9%

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

   8. Taylor expanded in kx around 0 88.9%

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

   if 4.9999999999999999e-17 < (sin.f64 kx)

   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. unpow2 99.6%

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

   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 55.2%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.015:\\ \;\;\;\;\left(\sin th \cdot ky\right) \cdot \sqrt{\frac{1}{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-59}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.025:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-57}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin ky \leq 10^{-16}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.025)
   (* (sin ky) (fabs (/ (sin th) (sin ky))))
   (if (<= (sin ky) 1e-57)
     (* (sin th) (/ (sin ky) (fabs (sin kx))))
     (if (<= (sin ky) 1e-16)
       (/ (* th ky) (hypot (sin ky) (sin kx)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.025) {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	} else if (sin(ky) <= 1e-57) {
		tmp = sin(th) * (sin(ky) / fabs(sin(kx)));
	} else if (sin(ky) <= 1e-16) {
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.025) {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	} else if (Math.sin(ky) <= 1e-57) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	} else if (Math.sin(ky) <= 1e-16) {
		tmp = (th * ky) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.025:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	elif math.sin(ky) <= 1e-57:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(kx)))
	elif math.sin(ky) <= 1e-16:
		tmp = (th * ky) / math.hypot(math.sin(ky), math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.025)
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	elseif (sin(ky) <= 1e-57)
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(kx))));
	elseif (sin(ky) <= 1e-16)
		tmp = Float64(Float64(th * ky) / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.025)
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	elseif (sin(ky) <= 1e-57)
		tmp = sin(th) * (sin(ky) / abs(sin(kx)));
	elseif (sin(ky) <= 1e-16)
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.025], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-57], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-16], N[(N[(th * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.025000000000000001

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.7%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.7%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 23.5%

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

      3. pow2 23.5%

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

   7. Applied egg-rr 23.5%

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

      1. unpow2 23.5%

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

      2. rem-sqrt-square 32.2%

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

   9. Simplified 32.2%

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

   if -0.025000000000000001 < (sin.f64 ky) < 9.99999999999999955e-58

   1. Initial program 85.6%

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

      1. unpow2 85.6%

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

      2. sin-mult 72.2%

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

   4. Applied egg-rr 72.2%

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

      1. div-sub 72.2%

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

      2. +-inverses 72.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      3. cos-0 72.2%

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

      4. metadata-eval 72.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{0.5} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      5. count-2 72.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot kx\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      6. *-commutative 72.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(kx \cdot 2\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

   6. Simplified 72.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \frac{\cos \left(kx \cdot 2\right)}{2}\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0 54.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 54.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)} \cdot \sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}}} \cdot \sin th \]

      2. rem-sqrt-square 54.3%

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}\right|}} \cdot \sin th \]

      3. sqr-sin-a 68.2%

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

      4. sqrt-unprod 40.9%

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

      5. add-sqr-sqrt 75.6%

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

   9. Applied egg-rr 75.6%

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

   if 9.99999999999999955e-58 < (sin.f64 ky) < 9.9999999999999998e-17

   1. Initial program 99.2%

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

      1. unpow2 99.2%

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

      2. sqr-neg 99.2%

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

      3. sin-neg 99.2%

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

      4. sin-neg 99.2%

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

      5. unpow2 99.2%

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

      6. associate-*l/ 99.0%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   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. associate-*r/ 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in th around 0 82.7%

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

   8. Taylor expanded in ky around 0 82.7%

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

   if 9.9999999999999998e-17 < (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/ 99.7%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   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 55.5%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.025:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-57}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin ky \leq 10^{-16}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.01)
   (/ (* th ky) (hypot (sin ky) (sin kx)))
   (if (<= (sin kx) 2e-51) (sin th) (* (sin ky) (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.01) {
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	} else if (sin(kx) <= 2e-51) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.01) {
		tmp = (th * ky) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else if (Math.sin(kx) <= 2e-51) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.01:
		tmp = (th * ky) / math.hypot(math.sin(ky), math.sin(kx))
	elif math.sin(kx) <= 2e-51:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.01)
		tmp = Float64(Float64(th * ky) / hypot(sin(ky), sin(kx)));
	elseif (sin(kx) <= 2e-51)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.01)
		tmp = (th * ky) / hypot(sin(ky), sin(kx));
	elseif (sin(kx) <= 2e-51)
		tmp = sin(th);
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.01], N[(N[(th * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-51], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0100000000000000002

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. sqr-neg 99.3%

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

      3. sin-neg 99.3%

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

      4. sin-neg 99.3%

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

      5. unpow2 99.3%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   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. associate-*r/ 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in th around 0 47.4%

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

   8. Taylor expanded in ky around 0 28.7%

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

   if -0.0100000000000000002 < (sin.f64 kx) < 2e-51

   1. Initial program 83.8%

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

      1. unpow2 83.8%

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

      2. sqr-neg 83.8%

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

      3. sin-neg 83.8%

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

      4. sin-neg 83.8%

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

      5. unpow2 83.8%

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

      6. associate-*l/ 79.2%

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

      7. associate-/l* 83.7%

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

      8. unpow2 83.7%

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

   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 32.8%

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

   if 2e-51 < (sin.f64 kx)

   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. unpow2 99.6%

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

   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 55.6%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 93.2%

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

   1. unpow2 93.2%

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

   2. sqr-neg 93.2%

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

   3. sin-neg 93.2%

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

   4. sin-neg 93.2%

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

   5. unpow2 93.2%

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

   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* 93.2%

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

   8. unpow2 93.2%

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

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. Final simplification 99.6%

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

Alternative 9: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.2%

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

   1. +-commutative 93.2%

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

   2. unpow2 93.2%

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

   3. unpow2 93.2%

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

   4. hypot-undefine 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 10: 68.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;kx \leq 1.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th \cdot \sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 3.6e-6)
   (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
   (if (<= kx 1.2e+82)
     (/ 1.0 (/ (hypot (sin ky) (sin kx)) (* th (sin ky))))
     (* (sin th) (/ (sin ky) (fabs (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.6e-6) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else if (kx <= 1.2e+82) {
		tmp = 1.0 / (hypot(sin(ky), sin(kx)) / (th * sin(ky)));
	} else {
		tmp = sin(th) * (sin(ky) / fabs(sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.6e-6) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else if (kx <= 1.2e+82) {
		tmp = 1.0 / (Math.hypot(Math.sin(ky), Math.sin(kx)) / (th * Math.sin(ky)));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 3.6e-6:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	elif kx <= 1.2e+82:
		tmp = 1.0 / (math.hypot(math.sin(ky), math.sin(kx)) / (th * math.sin(ky)))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 3.6e-6)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (kx <= 1.2e+82)
		tmp = Float64(1.0 / Float64(hypot(sin(ky), sin(kx)) / Float64(th * sin(ky))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 3.6e-6)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	elseif (kx <= 1.2e+82)
		tmp = 1.0 / (hypot(sin(ky), sin(kx)) / (th * sin(ky)));
	else
		tmp = sin(th) * (sin(ky) / abs(sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 3.6e-6], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 1.2e+82], N[(1.0 / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 3.59999999999999984e-6

   1. Initial program 91.0%

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

      1. +-commutative 91.0%

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

      2. unpow2 91.0%

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

      3. unpow2 91.0%

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

      4. hypot-undefine 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in kx around 0 66.6%

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

   if 3.59999999999999984e-6 < kx < 1.19999999999999999e82

   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/ 99.4%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   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/ 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in th around 0 61.8%

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

      1. clear-num 61.9%

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

      2. inv-pow 61.9%

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

      3. *-commutative 61.9%

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

   9. Applied egg-rr 61.9%

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

       1. unpow-1 61.9%

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

       2. *-commutative 61.9%

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

   11. Simplified 61.9%

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

   if 1.19999999999999999e82 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. sin-mult 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. div-sub 98.9%

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

      2. +-inverses 98.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      3. cos-0 98.9%

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

      4. metadata-eval 98.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{0.5} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      5. count-2 98.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot kx\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      6. *-commutative 98.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(kx \cdot 2\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

   6. Simplified 98.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \frac{\cos \left(kx \cdot 2\right)}{2}\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0 68.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 68.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)} \cdot \sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}}} \cdot \sin th \]

      2. rem-sqrt-square 68.6%

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}\right|}} \cdot \sin th \]

      3. sqr-sin-a 69.0%

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

      4. sqrt-unprod 31.2%

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

      5. add-sqr-sqrt 69.0%

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

   9. Applied egg-rr 69.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;kx \leq 1.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th \cdot \sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;kx \leq 1.6 \cdot 10^{+82}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 3.6e-6)
   (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
   (if (<= kx 1.6e+82)
     (* th (/ (sin ky) (hypot (sin ky) (sin kx))))
     (* (sin th) (/ (sin ky) (fabs (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.6e-6) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else if (kx <= 1.6e+82) {
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(th) * (sin(ky) / fabs(sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.6e-6) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else if (kx <= 1.6e+82) {
		tmp = th * (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 3.6e-6:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	elif kx <= 1.6e+82:
		tmp = th * (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 3.6e-6)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (kx <= 1.6e+82)
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 3.6e-6)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	elseif (kx <= 1.6e+82)
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(th) * (sin(ky) / abs(sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 3.6e-6], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 1.6e+82], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 3.59999999999999984e-6

   1. Initial program 91.0%

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

      1. +-commutative 91.0%

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

      2. unpow2 91.0%

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

      3. unpow2 91.0%

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

      4. hypot-undefine 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in kx around 0 66.6%

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

   if 3.59999999999999984e-6 < kx < 1.59999999999999987e82

   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/ 99.4%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   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/ 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in th around 0 61.8%

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

      1. associate-/l* 61.9%

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

      2. *-commutative 61.9%

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

   9. Applied egg-rr 61.9%

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

   if 1.59999999999999987e82 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. sin-mult 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. div-sub 98.9%

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

      2. +-inverses 98.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      3. cos-0 98.9%

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

      4. metadata-eval 98.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{0.5} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      5. count-2 98.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot kx\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      6. *-commutative 98.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(kx \cdot 2\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

   6. Simplified 98.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \frac{\cos \left(kx \cdot 2\right)}{2}\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0 68.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 68.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)} \cdot \sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}}} \cdot \sin th \]

      2. rem-sqrt-square 68.6%

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}\right|}} \cdot \sin th \]

      3. sqr-sin-a 69.0%

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

      4. sqrt-unprod 31.2%

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

      5. add-sqr-sqrt 69.0%

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

   9. Applied egg-rr 69.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;kx \leq 1.6 \cdot 10^{+82}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;kx \leq 3.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 3.6e-6)
   (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
   (if (<= kx 3.6e+81)
     (/ (* th (sin ky)) (hypot (sin ky) (sin kx)))
     (* (sin th) (/ (sin ky) (fabs (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.6e-6) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else if (kx <= 3.6e+81) {
		tmp = (th * sin(ky)) / hypot(sin(ky), sin(kx));
	} else {
		tmp = sin(th) * (sin(ky) / fabs(sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.6e-6) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else if (kx <= 3.6e+81) {
		tmp = (th * Math.sin(ky)) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 3.6e-6:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	elif kx <= 3.6e+81:
		tmp = (th * math.sin(ky)) / math.hypot(math.sin(ky), math.sin(kx))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 3.6e-6)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (kx <= 3.6e+81)
		tmp = Float64(Float64(th * sin(ky)) / hypot(sin(ky), sin(kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 3.6e-6)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	elseif (kx <= 3.6e+81)
		tmp = (th * sin(ky)) / hypot(sin(ky), sin(kx));
	else
		tmp = sin(th) * (sin(ky) / abs(sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 3.6e-6], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 3.6e+81], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 3.59999999999999984e-6

   1. Initial program 91.0%

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

      1. +-commutative 91.0%

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

      2. unpow2 91.0%

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

      3. unpow2 91.0%

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

      4. hypot-undefine 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in kx around 0 66.6%

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

   if 3.59999999999999984e-6 < kx < 3.60000000000000005e81

   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/ 99.4%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   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/ 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in th around 0 61.8%

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

   if 3.60000000000000005e81 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. sin-mult 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. div-sub 98.9%

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

      2. +-inverses 98.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      3. cos-0 98.9%

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

      4. metadata-eval 98.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{0.5} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      5. count-2 98.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot kx\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      6. *-commutative 98.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(kx \cdot 2\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

   6. Simplified 98.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \frac{\cos \left(kx \cdot 2\right)}{2}\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0 68.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 68.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)} \cdot \sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}}} \cdot \sin th \]

      2. rem-sqrt-square 68.6%

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}\right|}} \cdot \sin th \]

      3. sqr-sin-a 69.0%

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

      4. sqrt-unprod 31.2%

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

      5. add-sqr-sqrt 69.0%

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

   9. Applied egg-rr 69.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;kx \leq 3.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.102:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.102)
   (* (sin ky) (/ (sin th) (hypot (sin ky) kx)))
   (* (sin th) (/ (sin ky) (fabs (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.102) {
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	} else {
		tmp = sin(th) * (sin(ky) / fabs(sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.102) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(Math.sin(ky), kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 0.102:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(math.sin(ky), kx))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.102)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(sin(ky), kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 0.102)
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	else
		tmp = sin(th) * (sin(ky) / abs(sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.102], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.101999999999999993

   1. Initial program 91.1%

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

      1. unpow2 91.1%

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

      2. sqr-neg 91.1%

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

      3. sin-neg 91.1%

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

      4. sin-neg 91.1%

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

      5. unpow2 91.1%

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

      6. associate-*l/ 88.6%

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

      7. associate-/l* 91.0%

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

      8. unpow2 91.0%

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

   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 66.4%

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

   if 0.101999999999999993 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. sin-mult 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. div-sub 99.0%

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

      2. +-inverses 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      3. cos-0 99.0%

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

      4. metadata-eval 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{0.5} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      5. count-2 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot kx\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      6. *-commutative 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(kx \cdot 2\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

   6. Simplified 99.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \frac{\cos \left(kx \cdot 2\right)}{2}\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0 66.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 66.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)} \cdot \sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}}} \cdot \sin th \]

      2. rem-sqrt-square 66.2%

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}\right|}} \cdot \sin th \]

      3. sqr-sin-a 66.6%

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

      4. sqrt-unprod 33.0%

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

      5. add-sqr-sqrt 66.6%

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

   9. Applied egg-rr 66.6%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.102:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.105:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.105)
   (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
   (* (sin th) (/ (sin ky) (fabs (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.105) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else {
		tmp = sin(th) * (sin(ky) / fabs(sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.105) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 0.105:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.105)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 0.105)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	else
		tmp = sin(th) * (sin(ky) / abs(sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.105], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.104999999999999996

   1. Initial program 91.1%

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

      1. +-commutative 91.1%

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

      2. unpow2 91.1%

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

      3. unpow2 91.1%

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

      4. hypot-undefine 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in kx around 0 66.5%

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

   if 0.104999999999999996 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. sin-mult 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. div-sub 99.0%

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

      2. +-inverses 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      3. cos-0 99.0%

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

      4. metadata-eval 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{0.5} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      5. count-2 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot kx\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      6. *-commutative 99.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(kx \cdot 2\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

   6. Simplified 99.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \frac{\cos \left(kx \cdot 2\right)}{2}\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0 66.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 66.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)} \cdot \sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}}} \cdot \sin th \]

      2. rem-sqrt-square 66.2%

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}\right|}} \cdot \sin th \]

      3. sqr-sin-a 66.6%

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

      4. sqrt-unprod 33.0%

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

      5. add-sqr-sqrt 66.6%

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

   9. Applied egg-rr 66.6%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.105:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 2.15 \cdot 10^{-47}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 2.15e-47) (sin th) (* (sin ky) (/ (sin th) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.15e-47) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 2.15d-47) then
        tmp = sin(th)
    else
        tmp = sin(ky) * (sin(th) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.15e-47) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 2.15e-47:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 2.15e-47)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 2.15e-47)
		tmp = sin(th);
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 2.15e-47], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 2.1499999999999999e-47

   1. Initial program 90.4%

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

      1. unpow2 90.4%

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

      2. sqr-neg 90.4%

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

      3. sin-neg 90.4%

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

      4. sin-neg 90.4%

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

      5. unpow2 90.4%

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

      6. associate-*l/ 87.7%

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

      7. associate-/l* 90.3%

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

      8. unpow2 90.3%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 23.2%

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

   if 2.1499999999999999e-47 < 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/ 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. unpow2 99.6%

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

   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 44.3%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 2.15 \cdot 10^{-47}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 9.6 \cdot 10^{-40}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 9.6e-40) (sin th) (* ky (/ (sin th) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 9.6e-40) {
		tmp = sin(th);
	} else {
		tmp = ky * (sin(th) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 9.6d-40) then
        tmp = sin(th)
    else
        tmp = ky * (sin(th) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 9.6e-40) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 9.6e-40:
		tmp = math.sin(th)
	else:
		tmp = ky * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 9.6e-40)
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 9.6e-40)
		tmp = sin(th);
	else
		tmp = ky * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 9.6e-40], N[Sin[th], $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 9.59999999999999965e-40

   1. Initial program 90.5%

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

      1. unpow2 90.5%

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

      2. sqr-neg 90.5%

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

      3. sin-neg 90.5%

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

      4. sin-neg 90.5%

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

      5. unpow2 90.5%

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

      6. associate-*l/ 87.8%

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

      7. associate-/l* 90.4%

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

      8. unpow2 90.4%

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

   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 23.6%

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

   if 9.59999999999999965e-40 < 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/ 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. unpow2 99.6%

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

   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 40.2%

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

      1. associate-/l* 40.2%

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

   7. Simplified 40.2%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 9.6 \cdot 10^{-40}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.45 \cdot 10^{-39}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.45e-39) (sin th) (* ky (/ th (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.45e-39) {
		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 <= 1.45d-39) 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 <= 1.45e-39) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (th / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.45e-39:
		tmp = math.sin(th)
	else:
		tmp = ky * (th / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.45e-39)
		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 <= 1.45e-39)
		tmp = sin(th);
	else
		tmp = ky * (th / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.45e-39], N[Sin[th], $MachinePrecision], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.44999999999999994e-39

   1. Initial program 90.5%

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

      1. unpow2 90.5%

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

      2. sqr-neg 90.5%

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

      3. sin-neg 90.5%

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

      4. sin-neg 90.5%

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

      5. unpow2 90.5%

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

      6. associate-*l/ 87.8%

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

      7. associate-/l* 90.4%

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

      8. unpow2 90.4%

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

   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 23.6%

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

   if 1.44999999999999994e-39 < 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/ 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. unpow2 99.6%

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

   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%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in th around 0 57.0%

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

   8. Taylor expanded in ky around 0 28.6%

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

      1. associate-/l* 28.6%

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

   10. Simplified 28.6%

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

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.45 \cdot 10^{-39}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 23.3% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.2%

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

   1. unpow2 93.2%

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

   2. sqr-neg 93.2%

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

   3. sin-neg 93.2%

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

   4. sin-neg 93.2%

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

   5. unpow2 93.2%

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

   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* 93.2%

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

   8. unpow2 93.2%

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

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 20.0%

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

6. Final simplification 20.0%

   \[\leadsto \sin th \]
7. Add Preprocessing

Alternative 19: 13.6% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.2%

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

   1. unpow2 93.2%

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

   2. sqr-neg 93.2%

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

   3. sin-neg 93.2%

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

   4. sin-neg 93.2%

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

   5. unpow2 93.2%

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

   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* 93.2%

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

   8. unpow2 93.2%

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

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 20.0%

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

6. Taylor expanded in th around 0 13.2%

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

7. Final simplification 13.2%

   \[\leadsto th \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(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)))