Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.6% → 99.7%

Time: 26.6s

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: 93.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

   1. remove-double-neg 95.0%

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

   2. sin-neg 95.0%

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

   3. unpow1 95.0%

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

   4. metadata-eval 95.0%

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

   5. sqr-pow 48.8%

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

   6. distribute-rgt-neg-in 48.8%

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

   7. associate-/l* 48.8%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 77.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.15)
   (/ (* ky (- (sin th))) (sin kx))
   (if (<= (sin kx) 4e-7)
     (* (sin ky) (/ (sin th) (hypot (sin ky) kx)))
     (* (sin th) (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.15) {
		tmp = (ky * -sin(th)) / sin(kx);
	} else if (sin(kx) <= 4e-7) {
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.15) {
		tmp = (ky * -Math.sin(th)) / Math.sin(kx);
	} else if (Math.sin(kx) <= 4e-7) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(Math.sin(ky), kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.15:
		tmp = (ky * -math.sin(th)) / math.sin(kx)
	elif math.sin(kx) <= 4e-7:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(math.sin(ky), kx))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.15)
		tmp = Float64(Float64(ky * Float64(-sin(th))) / sin(kx));
	elseif (sin(kx) <= 4e-7)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(sin(ky), kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.15)
		tmp = (ky * -sin(th)) / sin(kx);
	elseif (sin(kx) <= 4e-7)
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.15], N[(N[(ky * (-N[Sin[th], $MachinePrecision])), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-7], 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[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.149999999999999994

   1. Initial program 99.5%

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

      1. associate-*l/ 99.3%

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

      2. associate-*r/ 99.4%

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

      3. +-commutative 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r/ 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 10.8%

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

      1. add-sqr-sqrt 9.5%

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

      2. sqrt-unprod 16.8%

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

      3. pow2 16.8%

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

      4. associate-/r/ 16.8%

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

      5. *-commutative 16.8%

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

   8. Applied egg-rr 16.8%

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

   9. Taylor expanded in ky around -inf 51.7%

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

   if -0.149999999999999994 < (sin.f64 kx) < 3.9999999999999998e-7

   1. Initial program 90.8%

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

      1. associate-*l/ 84.9%

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

      2. associate-*r/ 90.7%

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

      3. +-commutative 90.7%

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

      4. unpow2 90.7%

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

      5. unpow2 90.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 96.9%

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

   if 3.9999999999999998e-7 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. remove-double-neg 99.5%

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

      2. sin-neg 99.5%

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

      3. unpow1 99.5%

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

      4. metadata-eval 99.5%

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

      5. sqr-pow 44.1%

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

      6. distribute-rgt-neg-in 44.1%

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

      7. associate-/l* 43.9%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 65.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 3: 77.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.15)
   (/ (* ky (- (sin th))) (sin kx))
   (if (<= (sin kx) 4e-7)
     (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
     (* (sin th) (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.15) {
		tmp = (ky * -sin(th)) / sin(kx);
	} else if (sin(kx) <= 4e-7) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.15) {
		tmp = (ky * -Math.sin(th)) / Math.sin(kx);
	} else if (Math.sin(kx) <= 4e-7) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.15:
		tmp = (ky * -math.sin(th)) / math.sin(kx)
	elif math.sin(kx) <= 4e-7:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.15)
		tmp = Float64(Float64(ky * Float64(-sin(th))) / sin(kx));
	elseif (sin(kx) <= 4e-7)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.15)
		tmp = (ky * -sin(th)) / sin(kx);
	elseif (sin(kx) <= 4e-7)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.15], N[(N[(ky * (-N[Sin[th], $MachinePrecision])), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-7], 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[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.149999999999999994

   1. Initial program 99.5%

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

      1. associate-*l/ 99.3%

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

      2. associate-*r/ 99.4%

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

      3. +-commutative 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r/ 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 10.8%

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

      1. add-sqr-sqrt 9.5%

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

      2. sqrt-unprod 16.8%

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

      3. pow2 16.8%

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

      4. associate-/r/ 16.8%

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

      5. *-commutative 16.8%

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

   8. Applied egg-rr 16.8%

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

   9. Taylor expanded in ky around -inf 51.7%

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

   if -0.149999999999999994 < (sin.f64 kx) < 3.9999999999999998e-7

   1. Initial program 90.8%

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

      1. remove-double-neg 90.8%

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

      2. sin-neg 90.8%

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

      3. unpow1 90.8%

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

      4. metadata-eval 90.8%

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

      5. sqr-pow 47.7%

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

      6. distribute-rgt-neg-in 47.7%

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

      7. associate-/l* 47.7%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 97.0%

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

   if 3.9999999999999998e-7 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. remove-double-neg 99.5%

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

      2. sin-neg 99.5%

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

      3. unpow1 99.5%

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

      4. metadata-eval 99.5%

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

      5. sqr-pow 44.1%

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

      6. distribute-rgt-neg-in 44.1%

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

      7. associate-/l* 43.9%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 65.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 4: 77.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.15)
   (/ (sin th) (/ (hypot ky (sin kx)) (sin ky)))
   (if (<= (sin kx) 4e-7)
     (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
     (* (sin th) (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.15) {
		tmp = sin(th) / (hypot(ky, sin(kx)) / sin(ky));
	} else if (sin(kx) <= 4e-7) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.15) {
		tmp = Math.sin(th) / (Math.hypot(ky, Math.sin(kx)) / Math.sin(ky));
	} else if (Math.sin(kx) <= 4e-7) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.15:
		tmp = math.sin(th) / (math.hypot(ky, math.sin(kx)) / math.sin(ky))
	elif math.sin(kx) <= 4e-7:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.15)
		tmp = Float64(sin(th) / Float64(hypot(ky, sin(kx)) / sin(ky)));
	elseif (sin(kx) <= 4e-7)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.15)
		tmp = sin(th) / (hypot(ky, sin(kx)) / sin(ky));
	elseif (sin(kx) <= 4e-7)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.15], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-7], 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[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.149999999999999994

   1. Initial program 99.5%

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

      1. associate-*l/ 99.3%

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

      2. associate-*r/ 99.4%

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

      3. +-commutative 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r/ 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 53.1%

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

   if -0.149999999999999994 < (sin.f64 kx) < 3.9999999999999998e-7

   1. Initial program 90.8%

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

      1. remove-double-neg 90.8%

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

      2. sin-neg 90.8%

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

      3. unpow1 90.8%

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

      4. metadata-eval 90.8%

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

      5. sqr-pow 47.7%

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

      6. distribute-rgt-neg-in 47.7%

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

      7. associate-/l* 47.7%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 97.0%

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

   if 3.9999999999999998e-7 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. remove-double-neg 99.5%

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

      2. sin-neg 99.5%

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

      3. unpow1 99.5%

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

      4. metadata-eval 99.5%

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

      5. sqr-pow 44.1%

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

      6. distribute-rgt-neg-in 44.1%

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

      7. associate-/l* 43.9%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 65.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 5: 65.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.15)
   (/ (* ky (- (sin th))) (sin kx))
   (if (<= (sin kx) 0.01)
     (* (sin th) (/ ky (hypot ky kx)))
     (* (sin ky) (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.15) {
		tmp = (ky * -sin(th)) / sin(kx);
	} else if (sin(kx) <= 0.01) {
		tmp = sin(th) * (ky / hypot(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) <= -0.15) {
		tmp = (ky * -Math.sin(th)) / Math.sin(kx);
	} else if (Math.sin(kx) <= 0.01) {
		tmp = Math.sin(th) * (ky / Math.hypot(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) <= -0.15:
		tmp = (ky * -math.sin(th)) / math.sin(kx)
	elif math.sin(kx) <= 0.01:
		tmp = math.sin(th) * (ky / math.hypot(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) <= -0.15)
		tmp = Float64(Float64(ky * Float64(-sin(th))) / sin(kx));
	elseif (sin(kx) <= 0.01)
		tmp = Float64(sin(th) * Float64(ky / hypot(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) <= -0.15)
		tmp = (ky * -sin(th)) / sin(kx);
	elseif (sin(kx) <= 0.01)
		tmp = sin(th) * (ky / hypot(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], -0.15], N[(N[(ky * (-N[Sin[th], $MachinePrecision])), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.01], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.149999999999999994

   1. Initial program 99.5%

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

      1. associate-*l/ 99.3%

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

      2. associate-*r/ 99.4%

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

      3. +-commutative 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r/ 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 10.8%

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

      1. add-sqr-sqrt 9.5%

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

      2. sqrt-unprod 16.8%

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

      3. pow2 16.8%

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

      4. associate-/r/ 16.8%

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

      5. *-commutative 16.8%

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

   8. Applied egg-rr 16.8%

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

   9. Taylor expanded in ky around -inf 51.7%

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

   if -0.149999999999999994 < (sin.f64 kx) < 0.0100000000000000002

   1. Initial program 90.8%

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

      1. remove-double-neg 90.8%

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

      2. sin-neg 90.8%

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

      3. unpow1 90.8%

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

      4. metadata-eval 90.8%

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

      5. sqr-pow 48.1%

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

      6. distribute-rgt-neg-in 48.1%

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

      7. associate-/l* 48.1%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 96.3%

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

   5. Taylor expanded in ky around 0 50.3%

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

   6. Taylor expanded in ky around 0 70.1%

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

   if 0.0100000000000000002 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. associate-*l/ 99.4%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. unpow2 99.5%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 66.1%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]

Alternative 6: 65.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.15)
   (/ (* ky (- (sin th))) (sin kx))
   (if (<= (sin kx) 0.01)
     (* (sin th) (/ ky (hypot ky kx)))
     (* (sin th) (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.15) {
		tmp = (ky * -sin(th)) / sin(kx);
	} else if (sin(kx) <= 0.01) {
		tmp = sin(th) * (ky / hypot(ky, kx));
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.15) {
		tmp = (ky * -Math.sin(th)) / Math.sin(kx);
	} else if (Math.sin(kx) <= 0.01) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.15:
		tmp = (ky * -math.sin(th)) / math.sin(kx)
	elif math.sin(kx) <= 0.01:
		tmp = math.sin(th) * (ky / math.hypot(ky, kx))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.15)
		tmp = Float64(Float64(ky * Float64(-sin(th))) / sin(kx));
	elseif (sin(kx) <= 0.01)
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.15)
		tmp = (ky * -sin(th)) / sin(kx);
	elseif (sin(kx) <= 0.01)
		tmp = sin(th) * (ky / hypot(ky, kx));
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.15], N[(N[(ky * (-N[Sin[th], $MachinePrecision])), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.01], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.149999999999999994

   1. Initial program 99.5%

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

      1. associate-*l/ 99.3%

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

      2. associate-*r/ 99.4%

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

      3. +-commutative 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r/ 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 10.8%

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

      1. add-sqr-sqrt 9.5%

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

      2. sqrt-unprod 16.8%

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

      3. pow2 16.8%

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

      4. associate-/r/ 16.8%

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

      5. *-commutative 16.8%

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

   8. Applied egg-rr 16.8%

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

   9. Taylor expanded in ky around -inf 51.7%

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

   if -0.149999999999999994 < (sin.f64 kx) < 0.0100000000000000002

   1. Initial program 90.8%

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

      1. remove-double-neg 90.8%

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

      2. sin-neg 90.8%

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

      3. unpow1 90.8%

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

      4. metadata-eval 90.8%

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

      5. sqr-pow 48.1%

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

      6. distribute-rgt-neg-in 48.1%

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

      7. associate-/l* 48.1%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 96.3%

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

   5. Taylor expanded in ky around 0 50.3%

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

   6. Taylor expanded in ky around 0 70.1%

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

   if 0.0100000000000000002 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. remove-double-neg 99.4%

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

      2. sin-neg 99.4%

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

      3. unpow1 99.4%

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

      4. metadata-eval 99.4%

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

      5. sqr-pow 43.0%

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

      6. distribute-rgt-neg-in 43.0%

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

      7. associate-/l* 42.9%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 66.1%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 7: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

   1. associate-*l/ 91.9%

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

   2. associate-*r/ 95.0%

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

   3. +-commutative 95.0%

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

   4. unpow2 95.0%

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

   5. unpow2 95.0%

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

   6. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 8: 75.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := \mathsf{hypot}\left(\sin ky, kx\right)\\ \mathbf{if}\;ky \leq -0.0014:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 0.0112:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;ky \leq 6 \cdot 10^{+64}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{t_2}\\ \mathbf{elif}\;ky \leq 6 \cdot 10^{+232}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* (sin ky) th) (hypot (sin ky) (sin kx))))
        (t_2 (hypot (sin ky) kx)))
   (if (<= ky -0.0014)
     t_1
     (if (<= ky 0.0112)
       (/ (sin th) (/ (hypot ky (sin kx)) (sin ky)))
       (if (<= ky 6e+64)
         (* (sin ky) (/ (sin th) t_2))
         (if (<= ky 6e+232) t_1 (* (sin th) (/ (sin ky) t_2))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double t_2 = hypot(sin(ky), kx);
	double tmp;
	if (ky <= -0.0014) {
		tmp = t_1;
	} else if (ky <= 0.0112) {
		tmp = sin(th) / (hypot(ky, sin(kx)) / sin(ky));
	} else if (ky <= 6e+64) {
		tmp = sin(ky) * (sin(th) / t_2);
	} else if (ky <= 6e+232) {
		tmp = t_1;
	} else {
		tmp = sin(th) * (sin(ky) / t_2);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) * th) / Math.hypot(Math.sin(ky), Math.sin(kx));
	double t_2 = Math.hypot(Math.sin(ky), kx);
	double tmp;
	if (ky <= -0.0014) {
		tmp = t_1;
	} else if (ky <= 0.0112) {
		tmp = Math.sin(th) / (Math.hypot(ky, Math.sin(kx)) / Math.sin(ky));
	} else if (ky <= 6e+64) {
		tmp = Math.sin(ky) * (Math.sin(th) / t_2);
	} else if (ky <= 6e+232) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / t_2);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) * th) / math.hypot(math.sin(ky), math.sin(kx))
	t_2 = math.hypot(math.sin(ky), kx)
	tmp = 0
	if ky <= -0.0014:
		tmp = t_1
	elif ky <= 0.0112:
		tmp = math.sin(th) / (math.hypot(ky, math.sin(kx)) / math.sin(ky))
	elif ky <= 6e+64:
		tmp = math.sin(ky) * (math.sin(th) / t_2)
	elif ky <= 6e+232:
		tmp = t_1
	else:
		tmp = math.sin(th) * (math.sin(ky) / t_2)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	t_2 = hypot(sin(ky), kx)
	tmp = 0.0
	if (ky <= -0.0014)
		tmp = t_1;
	elseif (ky <= 0.0112)
		tmp = Float64(sin(th) / Float64(hypot(ky, sin(kx)) / sin(ky)));
	elseif (ky <= 6e+64)
		tmp = Float64(sin(ky) * Float64(sin(th) / t_2));
	elseif (ky <= 6e+232)
		tmp = t_1;
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	t_2 = hypot(sin(ky), kx);
	tmp = 0.0;
	if (ky <= -0.0014)
		tmp = t_1;
	elseif (ky <= 0.0112)
		tmp = sin(th) / (hypot(ky, sin(kx)) / sin(ky));
	elseif (ky <= 6e+64)
		tmp = sin(ky) * (sin(th) / t_2);
	elseif (ky <= 6e+232)
		tmp = t_1;
	else
		tmp = sin(th) * (sin(ky) / t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]}, If[LessEqual[ky, -0.0014], t$95$1, If[LessEqual[ky, 0.0112], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 6e+64], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 6e+232], t$95$1, N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if ky < -0.00139999999999999999 or 6.0000000000000004e64 < ky < 6.00000000000000006e232

   1. Initial program 99.6%

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

      1. associate-*l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 65.1%

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

   if -0.00139999999999999999 < ky < 0.0111999999999999999

   1. Initial program 90.4%

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

      1. associate-*l/ 84.4%

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

      2. associate-*r/ 90.3%

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

      3. +-commutative 90.3%

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

      4. unpow2 90.3%

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

      5. unpow2 90.3%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in ky around 0 99.6%

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

   if 0.0111999999999999999 < ky < 6.0000000000000004e64

   1. Initial program 100.0%

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

      1. associate-*l/ 99.6%

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

      2. associate-*r/ 100.0%

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

      3. +-commutative 100.0%

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

      4. unpow2 100.0%

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

      5. unpow2 100.0%

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

      6. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 100.0%

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

   if 6.00000000000000006e232 < ky

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. sin-neg 99.8%

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

      3. unpow1 99.8%

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

      4. metadata-eval 99.8%

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

      5. sqr-pow 54.6%

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

      6. distribute-rgt-neg-in 54.6%

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

      7. associate-/l* 54.5%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 57.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -0.0014:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;ky \leq 0.0112:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;ky \leq 6 \cdot 10^{+64}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;ky \leq 6 \cdot 10^{+232}:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \end{array} \]

Alternative 9: 62.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\sin th \cdot \frac{-ky}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.15)
   (* (sin th) (/ (- ky) (sin kx)))
   (if (<= (sin kx) 0.01)
     (* (sin th) (/ ky (hypot ky kx)))
     (* (sin th) (/ ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.15) {
		tmp = sin(th) * (-ky / sin(kx));
	} else if (sin(kx) <= 0.01) {
		tmp = sin(th) * (ky / hypot(ky, kx));
	} else {
		tmp = sin(th) * (ky / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.15) {
		tmp = Math.sin(th) * (-ky / Math.sin(kx));
	} else if (Math.sin(kx) <= 0.01) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, kx));
	} else {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.15:
		tmp = math.sin(th) * (-ky / math.sin(kx))
	elif math.sin(kx) <= 0.01:
		tmp = math.sin(th) * (ky / math.hypot(ky, kx))
	else:
		tmp = math.sin(th) * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.15)
		tmp = Float64(sin(th) * Float64(Float64(-ky) / sin(kx)));
	elseif (sin(kx) <= 0.01)
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, kx)));
	else
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.15)
		tmp = sin(th) * (-ky / sin(kx));
	elseif (sin(kx) <= 0.01)
		tmp = sin(th) * (ky / hypot(ky, kx));
	else
		tmp = sin(th) * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.15], N[(N[Sin[th], $MachinePrecision] * N[((-ky) / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.01], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.149999999999999994

   1. Initial program 99.5%

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

      1. associate-*l/ 99.3%

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

      2. associate-*r/ 99.4%

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

      3. +-commutative 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r/ 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 10.8%

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

      1. add-sqr-sqrt 9.5%

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

      2. sqrt-unprod 16.8%

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

      3. pow2 16.8%

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

      4. associate-/r/ 16.8%

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

      5. *-commutative 16.8%

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

   8. Applied egg-rr 16.8%

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

   9. Taylor expanded in ky around -inf 51.7%

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

       1. *-commutative 51.7%

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

       2. associate-/l* 51.6%

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

       3. mul-1-neg 51.6%

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

       4. associate-/l* 51.7%

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

       5. associate-*r/ 51.7%

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

   11. Simplified 51.7%

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

   if -0.149999999999999994 < (sin.f64 kx) < 0.0100000000000000002

   1. Initial program 90.8%

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

      1. remove-double-neg 90.8%

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

      2. sin-neg 90.8%

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

      3. unpow1 90.8%

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

      4. metadata-eval 90.8%

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

      5. sqr-pow 48.1%

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

      6. distribute-rgt-neg-in 48.1%

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

      7. associate-/l* 48.1%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 96.3%

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

   5. Taylor expanded in ky around 0 50.3%

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

   6. Taylor expanded in ky around 0 70.1%

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

   if 0.0100000000000000002 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. associate-*l/ 99.4%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. unpow2 99.5%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 57.9%

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

      1. associate-/l* 57.8%

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

      2. associate-/r/ 58.0%

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

   6. Simplified 58.0%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\sin th \cdot \frac{-ky}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 10: 62.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{-\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin kx \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.15)
   (/ (- (sin th)) (/ (sin kx) ky))
   (if (<= (sin kx) 0.01)
     (* (sin th) (/ ky (hypot ky kx)))
     (* (sin th) (/ ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.15) {
		tmp = -sin(th) / (sin(kx) / ky);
	} else if (sin(kx) <= 0.01) {
		tmp = sin(th) * (ky / hypot(ky, kx));
	} else {
		tmp = sin(th) * (ky / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.15) {
		tmp = -Math.sin(th) / (Math.sin(kx) / ky);
	} else if (Math.sin(kx) <= 0.01) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, kx));
	} else {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.15:
		tmp = -math.sin(th) / (math.sin(kx) / ky)
	elif math.sin(kx) <= 0.01:
		tmp = math.sin(th) * (ky / math.hypot(ky, kx))
	else:
		tmp = math.sin(th) * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.15)
		tmp = Float64(Float64(-sin(th)) / Float64(sin(kx) / ky));
	elseif (sin(kx) <= 0.01)
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, kx)));
	else
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.15)
		tmp = -sin(th) / (sin(kx) / ky);
	elseif (sin(kx) <= 0.01)
		tmp = sin(th) * (ky / hypot(ky, kx));
	else
		tmp = sin(th) * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.15], N[((-N[Sin[th], $MachinePrecision]) / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.01], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.149999999999999994

   1. Initial program 99.5%

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

      1. associate-*l/ 99.3%

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

      2. associate-*r/ 99.4%

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

      3. +-commutative 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r/ 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 10.8%

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

      1. add-sqr-sqrt 9.5%

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

      2. sqrt-unprod 16.8%

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

      3. pow2 16.8%

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

      4. associate-/r/ 16.8%

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

      5. *-commutative 16.8%

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

   8. Applied egg-rr 16.8%

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

   9. Taylor expanded in ky around -inf 51.7%

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

       1. *-commutative 51.7%

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

       2. associate-/l* 51.6%

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

       3. associate-*r/ 51.6%

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

       4. neg-mul-1 51.6%

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

   11. Simplified 51.6%

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

   if -0.149999999999999994 < (sin.f64 kx) < 0.0100000000000000002

   1. Initial program 90.8%

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

      1. remove-double-neg 90.8%

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

      2. sin-neg 90.8%

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

      3. unpow1 90.8%

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

      4. metadata-eval 90.8%

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

      5. sqr-pow 48.1%

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

      6. distribute-rgt-neg-in 48.1%

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

      7. associate-/l* 48.1%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 96.3%

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

   5. Taylor expanded in ky around 0 50.3%

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

   6. Taylor expanded in ky around 0 70.1%

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

   if 0.0100000000000000002 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. associate-*l/ 99.4%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. unpow2 99.5%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 57.9%

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

      1. associate-/l* 57.8%

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

      2. associate-/r/ 58.0%

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

   6. Simplified 58.0%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{-\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin kx \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 11: 62.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.15)
   (/ (* ky (- (sin th))) (sin kx))
   (if (<= (sin kx) 0.01)
     (* (sin th) (/ ky (hypot ky kx)))
     (* (sin th) (/ ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.15) {
		tmp = (ky * -sin(th)) / sin(kx);
	} else if (sin(kx) <= 0.01) {
		tmp = sin(th) * (ky / hypot(ky, kx));
	} else {
		tmp = sin(th) * (ky / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.15) {
		tmp = (ky * -Math.sin(th)) / Math.sin(kx);
	} else if (Math.sin(kx) <= 0.01) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, kx));
	} else {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.15:
		tmp = (ky * -math.sin(th)) / math.sin(kx)
	elif math.sin(kx) <= 0.01:
		tmp = math.sin(th) * (ky / math.hypot(ky, kx))
	else:
		tmp = math.sin(th) * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.15)
		tmp = Float64(Float64(ky * Float64(-sin(th))) / sin(kx));
	elseif (sin(kx) <= 0.01)
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, kx)));
	else
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.15)
		tmp = (ky * -sin(th)) / sin(kx);
	elseif (sin(kx) <= 0.01)
		tmp = sin(th) * (ky / hypot(ky, kx));
	else
		tmp = sin(th) * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.15], N[(N[(ky * (-N[Sin[th], $MachinePrecision])), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.01], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.149999999999999994

   1. Initial program 99.5%

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

      1. associate-*l/ 99.3%

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

      2. associate-*r/ 99.4%

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

      3. +-commutative 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r/ 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 10.8%

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

      1. add-sqr-sqrt 9.5%

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

      2. sqrt-unprod 16.8%

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

      3. pow2 16.8%

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

      4. associate-/r/ 16.8%

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

      5. *-commutative 16.8%

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

   8. Applied egg-rr 16.8%

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

   9. Taylor expanded in ky around -inf 51.7%

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

   if -0.149999999999999994 < (sin.f64 kx) < 0.0100000000000000002

   1. Initial program 90.8%

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

      1. remove-double-neg 90.8%

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

      2. sin-neg 90.8%

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

      3. unpow1 90.8%

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

      4. metadata-eval 90.8%

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

      5. sqr-pow 48.1%

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

      6. distribute-rgt-neg-in 48.1%

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

      7. associate-/l* 48.1%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 96.3%

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

   5. Taylor expanded in ky around 0 50.3%

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

   6. Taylor expanded in ky around 0 70.1%

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

   if 0.0100000000000000002 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. associate-*l/ 99.4%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. unpow2 99.5%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 57.9%

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

      1. associate-/l* 57.8%

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

      2. associate-/r/ 58.0%

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

   6. Simplified 58.0%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 12: 33.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-113}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-126}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -2e-113)
   (* th (/ ky (sin kx)))
   (if (<= (sin ky) 1e-126) (* (sin th) (/ ky kx)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -2e-113) {
		tmp = th * (ky / sin(kx));
	} else if (sin(ky) <= 1e-126) {
		tmp = sin(th) * (ky / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-2d-113)) then
        tmp = th * (ky / sin(kx))
    else if (sin(ky) <= 1d-126) then
        tmp = sin(th) * (ky / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -2e-113) {
		tmp = th * (ky / Math.sin(kx));
	} else if (Math.sin(ky) <= 1e-126) {
		tmp = Math.sin(th) * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -2e-113:
		tmp = th * (ky / math.sin(kx))
	elif math.sin(ky) <= 1e-126:
		tmp = math.sin(th) * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -2e-113)
		tmp = Float64(th * Float64(ky / sin(kx)));
	elseif (sin(ky) <= 1e-126)
		tmp = Float64(sin(th) * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -2e-113)
		tmp = th * (ky / sin(kx));
	elseif (sin(ky) <= 1e-126)
		tmp = sin(th) * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-113], N[(th * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-126], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1.99999999999999996e-113

   1. Initial program 99.7%

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

      1. associate-*l/ 98.5%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in ky around 0 10.9%

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

      1. add-sqr-sqrt 4.6%

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

      2. sqrt-unprod 8.0%

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

      3. pow2 8.0%

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

      4. associate-/r/ 8.0%

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

      5. *-commutative 8.0%

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

   8. Applied egg-rr 8.0%

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

   9. Taylor expanded in th around 0 7.1%

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

       1. associate-/l* 7.1%

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

       2. associate-/r/ 7.1%

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

   11. Simplified 7.1%

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

   if -1.99999999999999996e-113 < (sin.f64 ky) < 9.9999999999999995e-127

   1. Initial program 83.5%

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

      1. remove-double-neg 83.5%

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

      2. sin-neg 83.5%

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

      3. unpow1 83.5%

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

      4. metadata-eval 83.5%

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

      5. sqr-pow 41.6%

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

      6. distribute-rgt-neg-in 41.6%

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

      7. associate-/l* 41.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 61.5%

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

   5. Taylor expanded in ky around 0 31.9%

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

      1. associate-/l* 36.5%

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

      2. associate-/r/ 36.5%

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

   7. Simplified 36.5%

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

   if 9.9999999999999995e-127 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. associate-*l/ 97.5%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 56.4%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-113}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-126}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 33.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-115}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{elif}\;\sin ky \leq 10^{-126}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -2e-115)
   (/ ky (/ (sin kx) th))
   (if (<= (sin ky) 1e-126) (* (sin th) (/ ky kx)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -2e-115) {
		tmp = ky / (sin(kx) / th);
	} else if (sin(ky) <= 1e-126) {
		tmp = sin(th) * (ky / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-2d-115)) then
        tmp = ky / (sin(kx) / th)
    else if (sin(ky) <= 1d-126) then
        tmp = sin(th) * (ky / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -2e-115) {
		tmp = ky / (Math.sin(kx) / th);
	} else if (Math.sin(ky) <= 1e-126) {
		tmp = Math.sin(th) * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -2e-115:
		tmp = ky / (math.sin(kx) / th)
	elif math.sin(ky) <= 1e-126:
		tmp = math.sin(th) * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -2e-115)
		tmp = Float64(ky / Float64(sin(kx) / th));
	elseif (sin(ky) <= 1e-126)
		tmp = Float64(sin(th) * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -2e-115)
		tmp = ky / (sin(kx) / th);
	elseif (sin(ky) <= 1e-126)
		tmp = sin(th) * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-115], N[(ky / N[(N[Sin[kx], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-126], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -2.0000000000000001e-115

   1. Initial program 99.7%

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

      1. associate-*l/ 98.5%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in ky around 0 11.9%

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

   7. Taylor expanded in th around 0 8.1%

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

      1. associate-/l* 8.1%

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

   9. Simplified 8.1%

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

   if -2.0000000000000001e-115 < (sin.f64 ky) < 9.9999999999999995e-127

   1. Initial program 83.3%

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

      1. remove-double-neg 83.3%

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

      2. sin-neg 83.3%

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

      3. unpow1 83.3%

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

      4. metadata-eval 83.3%

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

      5. sqr-pow 40.8%

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

      6. distribute-rgt-neg-in 40.8%

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

      7. associate-/l* 40.8%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 61.0%

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

   5. Taylor expanded in ky around 0 31.0%

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

      1. associate-/l* 35.6%

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

      2. associate-/r/ 35.7%

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

   7. Simplified 35.7%

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

   if 9.9999999999999995e-127 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. associate-*l/ 97.5%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 56.4%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-115}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{elif}\;\sin ky \leq 10^{-126}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 40.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-117) (* (sin th) (/ ky (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-117) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 2d-117) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 2e-117) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 2e-117:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-117)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 2e-117)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-117], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 2.00000000000000006e-117

   1. Initial program 92.8%

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

      1. associate-*l/ 88.8%

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

      2. associate-*r/ 92.7%

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

      3. +-commutative 92.7%

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

      4. unpow2 92.7%

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

      5. unpow2 92.7%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 28.2%

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

      1. associate-/l* 30.2%

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

      2. associate-/r/ 30.2%

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

   6. Simplified 30.2%

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

   if 2.00000000000000006e-117 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. associate-*l/ 98.5%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 56.5%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15: 33.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -0.95 \lor \neg \left(ky \leq 1.1 \cdot 10^{-123}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -0.95) (not (<= ky 1.1e-123)))
   (sin th)
   (* (sin th) (/ ky kx))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -0.95) || !(ky <= 1.1e-123)) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (ky / 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 ((ky <= (-0.95d0)) .or. (.not. (ky <= 1.1d-123))) then
        tmp = sin(th)
    else
        tmp = sin(th) * (ky / kx)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -0.95) || !(ky <= 1.1e-123)) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (ky / kx);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -0.95) or not (ky <= 1.1e-123):
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (ky / kx)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -0.95) || !(ky <= 1.1e-123))
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(ky / kx));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -0.95) || ~((ky <= 1.1e-123)))
		tmp = sin(th);
	else
		tmp = sin(th) * (ky / kx);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -0.95], N[Not[LessEqual[ky, 1.1e-123]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -0.94999999999999996 or 1.10000000000000003e-123 < ky

   1. Initial program 99.7%

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

      1. associate-*l/ 98.4%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 32.8%

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

   if -0.94999999999999996 < ky < 1.10000000000000003e-123

   1. Initial program 87.8%

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

      1. remove-double-neg 87.8%

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

      2. sin-neg 87.8%

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

      3. unpow1 87.8%

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

      4. metadata-eval 87.8%

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

      5. sqr-pow 57.0%

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

      6. distribute-rgt-neg-in 57.0%

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

      7. associate-/l* 57.0%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 56.9%

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

   5. Taylor expanded in ky around 0 25.6%

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

      1. associate-/l* 28.9%

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

      2. associate-/r/ 28.9%

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

   7. Simplified 28.9%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -0.95 \lor \neg \left(ky \leq 1.1 \cdot 10^{-123}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \end{array} \]

Alternative 16: 30.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -0.95 \lor \neg \left(ky \leq 1.35 \cdot 10^{-126}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -0.95) (not (<= ky 1.35e-126))) (sin th) (/ ky (/ kx th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -0.95) || !(ky <= 1.35e-126)) {
		tmp = sin(th);
	} else {
		tmp = ky / (kx / th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((ky <= (-0.95d0)) .or. (.not. (ky <= 1.35d-126))) then
        tmp = sin(th)
    else
        tmp = ky / (kx / th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -0.95) || !(ky <= 1.35e-126)) {
		tmp = Math.sin(th);
	} else {
		tmp = ky / (kx / th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -0.95) or not (ky <= 1.35e-126):
		tmp = math.sin(th)
	else:
		tmp = ky / (kx / th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -0.95) || !(ky <= 1.35e-126))
		tmp = sin(th);
	else
		tmp = Float64(ky / Float64(kx / th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -0.95) || ~((ky <= 1.35e-126)))
		tmp = sin(th);
	else
		tmp = ky / (kx / th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -0.95], N[Not[LessEqual[ky, 1.35e-126]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(ky / N[(kx / th), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -0.94999999999999996 or 1.34999999999999998e-126 < ky

   1. Initial program 99.7%

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

      1. associate-*l/ 98.4%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 32.8%

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

   if -0.94999999999999996 < ky < 1.34999999999999998e-126

   1. Initial program 87.8%

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

      1. remove-double-neg 87.8%

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

      2. sin-neg 87.8%

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

      3. unpow1 87.8%

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

      4. metadata-eval 87.8%

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

      5. sqr-pow 57.0%

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

      6. distribute-rgt-neg-in 57.0%

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

      7. associate-/l* 57.0%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 56.9%

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

   5. Taylor expanded in ky around 0 25.6%

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

      1. associate-/l* 28.9%

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

      2. associate-/r/ 28.9%

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

   7. Simplified 28.9%

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

   8. Taylor expanded in th around 0 17.4%

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

      1. associate-/l* 20.7%

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

   10. Simplified 20.7%

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

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -0.95 \lor \neg \left(ky \leq 1.35 \cdot 10^{-126}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \end{array} \]

Alternative 17: 20.7% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.02 \cdot 10^{+40}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 2.8 \cdot 10^{-117}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.02e+40) th (if (<= ky 2.8e-117) (* th (/ ky kx)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.02e+40) {
		tmp = th;
	} else if (ky <= 2.8e-117) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-1.02d+40)) then
        tmp = th
    else if (ky <= 2.8d-117) then
        tmp = th * (ky / kx)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.02e+40) {
		tmp = th;
	} else if (ky <= 2.8e-117) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.02e+40:
		tmp = th
	elif ky <= 2.8e-117:
		tmp = th * (ky / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.02e+40)
		tmp = th;
	elseif (ky <= 2.8e-117)
		tmp = Float64(th * Float64(ky / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.02e+40)
		tmp = th;
	elseif (ky <= 2.8e-117)
		tmp = th * (ky / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.02e+40], th, If[LessEqual[ky, 2.8e-117], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.02e40 or 2.8e-117 < ky

   1. Initial program 99.7%

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

      1. associate-*l/ 98.9%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 33.4%

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

   5. Taylor expanded in th around 0 22.2%

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

   if -1.02e40 < ky < 2.8e-117

   1. Initial program 89.0%

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

      1. remove-double-neg 89.0%

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

      2. sin-neg 89.0%

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

      3. unpow1 89.0%

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

      4. metadata-eval 89.0%

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

      5. sqr-pow 56.7%

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

      6. distribute-rgt-neg-in 56.7%

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

      7. associate-/l* 56.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 57.7%

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

   5. Taylor expanded in ky around 0 23.4%

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

      1. associate-/l* 26.5%

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

      2. associate-/r/ 26.5%

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

   7. Simplified 26.5%

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

   8. Taylor expanded in th around 0 16.1%

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

      1. associate-/l* 19.1%

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

   10. Simplified 19.1%

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

       1. associate-/r/ 19.1%

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

   12. Applied egg-rr 19.1%

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

4. Final simplification 20.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.02 \cdot 10^{+40}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 2.8 \cdot 10^{-117}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 18: 20.7% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.02 \cdot 10^{+40}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 3.1 \cdot 10^{-117}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.02e+40) th (if (<= ky 3.1e-117) (/ ky (/ kx th)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.02e+40) {
		tmp = th;
	} else if (ky <= 3.1e-117) {
		tmp = ky / (kx / th);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-1.02d+40)) then
        tmp = th
    else if (ky <= 3.1d-117) then
        tmp = ky / (kx / th)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.02e+40) {
		tmp = th;
	} else if (ky <= 3.1e-117) {
		tmp = ky / (kx / th);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.02e+40:
		tmp = th
	elif ky <= 3.1e-117:
		tmp = ky / (kx / th)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.02e+40)
		tmp = th;
	elseif (ky <= 3.1e-117)
		tmp = Float64(ky / Float64(kx / th));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.02e+40)
		tmp = th;
	elseif (ky <= 3.1e-117)
		tmp = ky / (kx / th);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.02e+40], th, If[LessEqual[ky, 3.1e-117], N[(ky / N[(kx / th), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.02e40 or 3.10000000000000011e-117 < ky

   1. Initial program 99.7%

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

      1. associate-*l/ 98.9%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 33.4%

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

   5. Taylor expanded in th around 0 22.2%

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

   if -1.02e40 < ky < 3.10000000000000011e-117

   1. Initial program 89.0%

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

      1. remove-double-neg 89.0%

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

      2. sin-neg 89.0%

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

      3. unpow1 89.0%

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

      4. metadata-eval 89.0%

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

      5. sqr-pow 56.7%

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

      6. distribute-rgt-neg-in 56.7%

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

      7. associate-/l* 56.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 57.7%

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

   5. Taylor expanded in ky around 0 23.4%

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

      1. associate-/l* 26.5%

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

      2. associate-/r/ 26.5%

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

   7. Simplified 26.5%

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

   8. Taylor expanded in th around 0 16.1%

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

      1. associate-/l* 19.1%

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

   10. Simplified 19.1%

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

4. Final simplification 20.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.02 \cdot 10^{+40}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 3.1 \cdot 10^{-117}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 19: 13.1% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 95.0%

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

   1. associate-*l/ 91.9%

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

   2. associate-*r/ 95.0%

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

   3. +-commutative 95.0%

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

   4. unpow2 95.0%

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

   5. unpow2 95.0%

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

   6. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in kx around 0 22.1%

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

5. Taylor expanded in th around 0 14.8%

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

6. Final simplification 14.8%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023301 
(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)))