Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 22.5s

Alternatives: 16

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% 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 94.5%

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

   1. +-commutative 94.5%

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

   2. unpow2 94.5%

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

   3. unpow2 94.5%

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

   4. hypot-def 99.7%

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

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: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.7:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin th \leq -0.44:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq -0.05:\\ \;\;\;\;\sin ky \cdot \left(\left(1 + \frac{\sin th}{\sin kx}\right) + -1\right)\\ \mathbf{elif}\;\sin th \leq 0.0005:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.7)
   (/ (sin ky) (/ (sin kx) (sin th)))
   (if (<= (sin th) -0.44)
     (fabs (sin th))
     (if (<= (sin th) -0.05)
       (* (sin ky) (+ (+ 1.0 (/ (sin th) (sin kx))) -1.0))
       (if (<= (sin th) 0.0005)
         (/ (sin ky) (/ (hypot (sin ky) (sin kx)) th))
         (* (sin ky) (fabs (/ (sin th) (sin ky)))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.7) {
		tmp = sin(ky) / (sin(kx) / sin(th));
	} else if (sin(th) <= -0.44) {
		tmp = fabs(sin(th));
	} else if (sin(th) <= -0.05) {
		tmp = sin(ky) * ((1.0 + (sin(th) / sin(kx))) + -1.0);
	} else if (sin(th) <= 0.0005) {
		tmp = sin(ky) / (hypot(sin(ky), sin(kx)) / th);
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.7) {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	} else if (Math.sin(th) <= -0.44) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(th) <= -0.05) {
		tmp = Math.sin(ky) * ((1.0 + (Math.sin(th) / Math.sin(kx))) + -1.0);
	} else if (Math.sin(th) <= 0.0005) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(ky), Math.sin(kx)) / th);
	} else {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.7:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	elif math.sin(th) <= -0.44:
		tmp = math.fabs(math.sin(th))
	elif math.sin(th) <= -0.05:
		tmp = math.sin(ky) * ((1.0 + (math.sin(th) / math.sin(kx))) + -1.0)
	elif math.sin(th) <= 0.0005:
		tmp = math.sin(ky) / (math.hypot(math.sin(ky), math.sin(kx)) / th)
	else:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.7)
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	elseif (sin(th) <= -0.44)
		tmp = abs(sin(th));
	elseif (sin(th) <= -0.05)
		tmp = Float64(sin(ky) * Float64(Float64(1.0 + Float64(sin(th) / sin(kx))) + -1.0));
	elseif (sin(th) <= 0.0005)
		tmp = Float64(sin(ky) / Float64(hypot(sin(ky), sin(kx)) / th));
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.7)
		tmp = sin(ky) / (sin(kx) / sin(th));
	elseif (sin(th) <= -0.44)
		tmp = abs(sin(th));
	elseif (sin(th) <= -0.05)
		tmp = sin(ky) * ((1.0 + (sin(th) / sin(kx))) + -1.0);
	elseif (sin(th) <= 0.0005)
		tmp = sin(ky) / (hypot(sin(ky), sin(kx)) / th);
	else
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.7], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], -0.44], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], -0.05], N[(N[Sin[ky], $MachinePrecision] * N[(N[(1.0 + N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.0005], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 th) < -0.69999999999999996

   1. Initial program 98.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. *-commutative 98.9%

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

      3. associate-*l/ 98.6%

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

      4. +-commutative 98.6%

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

      5. unpow2 98.6%

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

      6. unpow2 98.6%

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

      7. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in ky around 0 18.9%

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

      1. *-commutative 18.9%

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

      2. clear-num 19.0%

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

      3. un-div-inv 19.0%

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

   6. Applied egg-rr 19.0%

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

   if -0.69999999999999996 < (sin.f64 th) < -0.440000000000000002

   1. Initial program 91.7%

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

      1. associate-*l/ 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*l/ 91.8%

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

      4. +-commutative 91.8%

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

      5. unpow2 91.8%

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

      6. unpow2 91.8%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 11.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 47.7%

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

      3. pow2 47.7%

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

      4. *-commutative 47.7%

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

      5. clear-num 47.7%

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

      6. un-div-inv 47.6%

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

   6. Applied egg-rr 47.6%

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

      1. unpow2 47.6%

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

      2. rem-sqrt-square 47.6%

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

      3. associate-/r/ 47.7%

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

      4. *-inverses 47.7%

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

      5. *-lft-identity 47.7%

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

   8. Simplified 47.7%

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

   if -0.440000000000000002 < (sin.f64 th) < -0.050000000000000003

   1. Initial program 94.8%

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

      1. associate-*l/ 95.0%

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

      2. *-commutative 95.0%

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

      3. associate-*l/ 94.8%

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

      4. +-commutative 94.8%

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

      5. unpow2 94.8%

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

      6. unpow2 94.8%

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

      7. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 18.2%

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

      1. expm1-log1p-u 8.2%

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

   6. Applied egg-rr 8.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin th}{\sin kx}\right)\right)} \cdot \sin ky \]
   7. Step-by-step derivation

      1. expm1-udef 8.3%

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

      2. log1p-udef 8.3%

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

      3. rem-exp-log 18.3%

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

   8. Applied egg-rr 18.3%

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

   if -0.050000000000000003 < (sin.f64 th) < 5.0000000000000001e-4

   1. Initial program 95.2%

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

      1. associate-*l/ 93.8%

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

      2. +-commutative 93.8%

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

      3. unpow2 93.8%

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

      4. unpow2 93.8%

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

      5. hypot-udef 95.1%

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

      6. associate-*l/ 99.7%

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

      7. expm1-log1p-u 99.7%

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

      8. expm1-udef 21.0%

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

   3. Applied egg-rr 21.0%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in th around 0 93.1%

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

      1. associate-*l/ 93.3%

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

      2. unpow2 93.3%

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

      3. unpow2 93.3%

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

      4. hypot-def 97.8%

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

      5. *-lft-identity 97.8%

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

      6. hypot-def 93.3%

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

      7. unpow2 93.3%

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

      8. unpow2 93.3%

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

      9. +-commutative 93.3%

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

      10. unpow2 93.3%

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

      11. unpow2 93.3%

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

      12. hypot-def 97.8%

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

   8. Simplified 97.8%

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

   if 5.0000000000000001e-4 < (sin.f64 th)

   1. Initial program 91.7%

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

      1. associate-*l/ 91.7%

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

      2. *-commutative 91.7%

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

      3. associate-*l/ 91.6%

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

      4. +-commutative 91.6%

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

      5. unpow2 91.6%

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

      6. unpow2 91.6%

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

      7. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 33.8%

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

      1. add-sqr-sqrt 32.9%

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

      2. sqrt-unprod 48.7%

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

      3. pow2 48.7%

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

   6. Applied egg-rr 48.7%

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

      1. unpow2 48.7%

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

      2. rem-sqrt-square 55.1%

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

   8. Simplified 55.1%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.7:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin th \leq -0.44:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq -0.05:\\ \;\;\;\;\sin ky \cdot \left(\left(1 + \frac{\sin th}{\sin kx}\right) + -1\right)\\ \mathbf{elif}\;\sin th \leq 0.0005:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \end{array} \]

Alternative 3: 73.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin th \leq -0.02:\\ \;\;\;\;\frac{ky \cdot \sin th}{t_1}\\ \mathbf{elif}\;\sin th \leq 0.0005:\\ \;\;\;\;\frac{\sin ky}{t_1 \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin th) -0.02)
     (/ (* ky (sin th)) t_1)
     (if (<= (sin th) 0.0005)
       (/ (sin ky) (* t_1 (+ (/ 1.0 th) (* th 0.16666666666666666))))
       (* (sin ky) (fabs (/ (sin th) (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(th) <= -0.02) {
		tmp = (ky * sin(th)) / t_1;
	} else if (sin(th) <= 0.0005) {
		tmp = sin(ky) / (t_1 * ((1.0 / th) + (th * 0.16666666666666666)));
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (Math.sin(th) <= -0.02) {
		tmp = (ky * Math.sin(th)) / t_1;
	} else if (Math.sin(th) <= 0.0005) {
		tmp = Math.sin(ky) / (t_1 * ((1.0 / th) + (th * 0.16666666666666666)));
	} else {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(th) <= -0.02:
		tmp = (ky * math.sin(th)) / t_1
	elif math.sin(th) <= 0.0005:
		tmp = math.sin(ky) / (t_1 * ((1.0 / th) + (th * 0.16666666666666666)))
	else:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(th) <= -0.02)
		tmp = Float64(Float64(ky * sin(th)) / t_1);
	elseif (sin(th) <= 0.0005)
		tmp = Float64(sin(ky) / Float64(t_1 * Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666))));
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(th) <= -0.02)
		tmp = (ky * sin(th)) / t_1;
	elseif (sin(th) <= 0.0005)
		tmp = sin(ky) / (t_1 * ((1.0 / th) + (th * 0.16666666666666666)));
	else
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[Sin[th], $MachinePrecision], -0.02], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.0005], N[(N[Sin[ky], $MachinePrecision] / N[(t$95$1 * N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 96.1%

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

      1. associate-*l/ 96.2%

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

      2. +-commutative 96.2%

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

      3. unpow2 96.2%

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

      4. unpow2 96.2%

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

      5. hypot-udef 99.7%

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

      6. associate-*l/ 99.6%

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

      7. expm1-log1p-u 99.4%

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

      8. expm1-udef 63.4%

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

   3. Applied egg-rr 63.3%

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

      1. expm1-def 99.3%

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

      2. expm1-log1p 99.4%

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

      3. associate-/l* 99.7%

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

      4. hypot-def 96.2%

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

      5. unpow2 96.2%

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

      6. unpow2 96.2%

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

      7. +-commutative 96.2%

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

      8. unpow2 96.2%

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

      9. unpow2 96.2%

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

      10. hypot-def 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in ky around 0 50.7%

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

   if -0.0200000000000000004 < (sin.f64 th) < 5.0000000000000001e-4

   1. Initial program 95.2%

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

      1. associate-*l/ 93.7%

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

      2. +-commutative 93.7%

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

      3. unpow2 93.7%

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

      4. unpow2 93.7%

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

      5. hypot-udef 95.1%

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

      6. associate-*l/ 99.7%

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

      7. expm1-log1p-u 99.7%

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

      8. expm1-udef 21.2%

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

   3. Applied egg-rr 21.2%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in th around 0 94.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right) + \frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   7. Step-by-step derivation

      1. +-commutative 94.6%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      2. *-commutative 94.6%

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \frac{1}{th}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      3. unpow2 94.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      4. unpow2 94.6%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      5. hypot-def 99.1%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      6. associate-*r/ 99.4%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot 1}{th}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      7. *-commutative 99.4%

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{1 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)}}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      8. associate-*l/ 99.1%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      9. associate-*r* 99.1%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right) + \color{blue}{\left(0.16666666666666666 \cdot th\right) \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   8. Simplified 99.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\frac{1}{th} + 0.16666666666666666 \cdot th\right)}} \]

   if 5.0000000000000001e-4 < (sin.f64 th)

   1. Initial program 91.7%

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

      1. associate-*l/ 91.7%

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

      2. *-commutative 91.7%

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

      3. associate-*l/ 91.6%

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

      4. +-commutative 91.6%

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

      5. unpow2 91.6%

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

      6. unpow2 91.6%

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

      7. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 33.8%

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

      1. add-sqr-sqrt 32.9%

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

      2. sqrt-unprod 48.7%

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

      3. pow2 48.7%

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

   6. Applied egg-rr 48.7%

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

      1. unpow2 48.7%

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

      2. rem-sqrt-square 55.1%

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

   8. Simplified 55.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.02:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.0005:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \end{array} \]

Alternative 4: 50.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-52}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-245}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-52)
   (fabs (sin th))
   (if (<= (sin ky) 5e-245)
     (/ (sin th) (/ (sin kx) ky))
     (if (<= (sin ky) 1e-104) (* (sin th) (fabs (/ ky (sin kx)))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-52) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-245) {
		tmp = sin(th) / (sin(kx) / ky);
	} else if (sin(ky) <= 1e-104) {
		tmp = sin(th) * fabs((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) <= (-1d-52)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-245) then
        tmp = sin(th) / (sin(kx) / ky)
    else if (sin(ky) <= 1d-104) then
        tmp = sin(th) * abs((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) <= -1e-52) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-245) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else if (Math.sin(ky) <= 1e-104) {
		tmp = Math.sin(th) * Math.abs((ky / Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-52:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-245:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	elif math.sin(ky) <= 1e-104:
		tmp = math.sin(th) * math.fabs((ky / math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-52)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-245)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	elseif (sin(ky) <= 1e-104)
		tmp = Float64(sin(th) * abs(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) <= -1e-52)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-245)
		tmp = sin(th) / (sin(kx) / ky);
	elseif (sin(ky) <= 1e-104)
		tmp = sin(th) * abs((ky / sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-52], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-245], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-104], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -1e-52

   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/ 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.8%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 29.2%

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

      3. pow2 29.2%

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

      4. *-commutative 29.2%

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

      5. clear-num 29.2%

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

      6. un-div-inv 29.2%

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

   6. Applied egg-rr 29.2%

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

      1. unpow2 29.2%

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

      2. rem-sqrt-square 36.4%

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

      3. associate-/r/ 36.5%

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

      4. *-inverses 36.5%

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

      5. *-lft-identity 36.5%

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

   8. Simplified 36.5%

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

   if -1e-52 < (sin.f64 ky) < 4.9999999999999997e-245

   1. Initial program 79.5%

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

      1. *-commutative 79.5%

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

      2. clear-num 79.4%

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

      3. +-commutative 79.4%

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

      4. unpow2 79.4%

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

      5. unpow2 79.4%

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

      6. hypot-udef 99.5%

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

      7. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in ky around 0 54.5%

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

   if 4.9999999999999997e-245 < (sin.f64 ky) < 9.99999999999999927e-105

   1. Initial program 93.1%

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

   2. Taylor expanded in ky around 0 58.9%

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

      1. associate-/l* 58.9%

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

      2. associate-/r/ 58.9%

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

   4. Simplified 58.9%

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

      1. add-sqr-sqrt 51.1%

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

      2. sqrt-unprod 66.5%

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

      3. pow2 66.5%

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

   6. Applied egg-rr 66.5%

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

      1. unpow2 66.5%

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

      2. rem-sqrt-square 90.4%

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

   8. Simplified 90.4%

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

   if 9.99999999999999927e-105 < (sin.f64 ky)

   1. Initial program 99.8%

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

   2. Taylor expanded in kx around 0 61.8%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-52}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-245}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5: 50.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-52}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-245}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-52)
   (fabs (sin th))
   (if (<= (sin ky) 5e-245)
     (/ (sin th) (/ (sin kx) ky))
     (if (<= (sin ky) 1e-104)
       (* (sin th) (fabs (/ ky (sin kx))))
       (/ (* (sin ky) (sin th)) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-52) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-245) {
		tmp = sin(th) / (sin(kx) / ky);
	} else if (sin(ky) <= 1e-104) {
		tmp = sin(th) * fabs((ky / sin(kx)));
	} else {
		tmp = (sin(ky) * sin(th)) / sin(ky);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-52)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-245) then
        tmp = sin(th) / (sin(kx) / ky)
    else if (sin(ky) <= 1d-104) then
        tmp = sin(th) * abs((ky / sin(kx)))
    else
        tmp = (sin(ky) * sin(th)) / sin(ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-52) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-245) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else if (Math.sin(ky) <= 1e-104) {
		tmp = Math.sin(th) * Math.abs((ky / Math.sin(kx)));
	} else {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sin(ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-52:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-245:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	elif math.sin(ky) <= 1e-104:
		tmp = math.sin(th) * math.fabs((ky / math.sin(kx)))
	else:
		tmp = (math.sin(ky) * math.sin(th)) / math.sin(ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-52)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-245)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	elseif (sin(ky) <= 1e-104)
		tmp = Float64(sin(th) * abs(Float64(ky / sin(kx))));
	else
		tmp = Float64(Float64(sin(ky) * sin(th)) / sin(ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-52)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-245)
		tmp = sin(th) / (sin(kx) / ky);
	elseif (sin(ky) <= 1e-104)
		tmp = sin(th) * abs((ky / sin(kx)));
	else
		tmp = (sin(ky) * sin(th)) / sin(ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-52], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-245], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-104], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -1e-52

   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/ 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.8%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 29.2%

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

      3. pow2 29.2%

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

      4. *-commutative 29.2%

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

      5. clear-num 29.2%

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

      6. un-div-inv 29.2%

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

   6. Applied egg-rr 29.2%

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

      1. unpow2 29.2%

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

      2. rem-sqrt-square 36.4%

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

      3. associate-/r/ 36.5%

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

      4. *-inverses 36.5%

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

      5. *-lft-identity 36.5%

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

   8. Simplified 36.5%

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

   if -1e-52 < (sin.f64 ky) < 4.9999999999999997e-245

   1. Initial program 79.5%

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

      1. *-commutative 79.5%

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

      2. clear-num 79.4%

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

      3. +-commutative 79.4%

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

      4. unpow2 79.4%

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

      5. unpow2 79.4%

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

      6. hypot-udef 99.5%

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

      7. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in ky around 0 54.5%

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

   if 4.9999999999999997e-245 < (sin.f64 ky) < 9.99999999999999927e-105

   1. Initial program 93.1%

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

   2. Taylor expanded in ky around 0 58.9%

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

      1. associate-/l* 58.9%

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

      2. associate-/r/ 58.9%

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

   4. Simplified 58.9%

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

      1. add-sqr-sqrt 51.1%

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

      2. sqrt-unprod 66.5%

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

      3. pow2 66.5%

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

   6. Applied egg-rr 66.5%

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

      1. unpow2 66.5%

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

      2. rem-sqrt-square 90.4%

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

   8. Simplified 90.4%

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

   if 9.99999999999999927e-105 < (sin.f64 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. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l/ 99.5%

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

      4. +-commutative 99.5%

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

      5. unpow2 99.5%

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

      6. unpow2 99.5%

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

      7. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*r/ 62.7%

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

   6. Applied egg-rr 62.7%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-52}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-245}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \end{array} \]

Alternative 6: 73.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin th \leq -0.02:\\ \;\;\;\;\frac{ky \cdot \sin th}{t_1}\\ \mathbf{elif}\;\sin th \leq 0.0005:\\ \;\;\;\;\frac{\sin ky}{\frac{t_1}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin th) -0.02)
     (/ (* ky (sin th)) t_1)
     (if (<= (sin th) 0.0005)
       (/ (sin ky) (/ t_1 th))
       (* (sin ky) (fabs (/ (sin th) (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(th) <= -0.02) {
		tmp = (ky * sin(th)) / t_1;
	} else if (sin(th) <= 0.0005) {
		tmp = sin(ky) / (t_1 / th);
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (Math.sin(th) <= -0.02) {
		tmp = (ky * Math.sin(th)) / t_1;
	} else if (Math.sin(th) <= 0.0005) {
		tmp = Math.sin(ky) / (t_1 / th);
	} else {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(th) <= -0.02:
		tmp = (ky * math.sin(th)) / t_1
	elif math.sin(th) <= 0.0005:
		tmp = math.sin(ky) / (t_1 / th)
	else:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(th) <= -0.02)
		tmp = Float64(Float64(ky * sin(th)) / t_1);
	elseif (sin(th) <= 0.0005)
		tmp = Float64(sin(ky) / Float64(t_1 / th));
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(th) <= -0.02)
		tmp = (ky * sin(th)) / t_1;
	elseif (sin(th) <= 0.0005)
		tmp = sin(ky) / (t_1 / th);
	else
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[Sin[th], $MachinePrecision], -0.02], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.0005], N[(N[Sin[ky], $MachinePrecision] / N[(t$95$1 / th), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 96.1%

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

      1. associate-*l/ 96.2%

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

      2. +-commutative 96.2%

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

      3. unpow2 96.2%

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

      4. unpow2 96.2%

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

      5. hypot-udef 99.7%

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

      6. associate-*l/ 99.6%

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

      7. expm1-log1p-u 99.4%

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

      8. expm1-udef 63.4%

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

   3. Applied egg-rr 63.3%

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

      1. expm1-def 99.3%

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

      2. expm1-log1p 99.4%

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

      3. associate-/l* 99.7%

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

      4. hypot-def 96.2%

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

      5. unpow2 96.2%

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

      6. unpow2 96.2%

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

      7. +-commutative 96.2%

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

      8. unpow2 96.2%

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

      9. unpow2 96.2%

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

      10. hypot-def 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in ky around 0 50.7%

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

   if -0.0200000000000000004 < (sin.f64 th) < 5.0000000000000001e-4

   1. Initial program 95.2%

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

      1. associate-*l/ 93.7%

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

      2. +-commutative 93.7%

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

      3. unpow2 93.7%

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

      4. unpow2 93.7%

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

      5. hypot-udef 95.1%

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

      6. associate-*l/ 99.7%

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

      7. expm1-log1p-u 99.7%

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

      8. expm1-udef 21.2%

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

   3. Applied egg-rr 21.2%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in th around 0 93.8%

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

      1. associate-*l/ 94.0%

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

      2. unpow2 94.0%

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

      3. unpow2 94.0%

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

      4. hypot-def 98.5%

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

      5. *-lft-identity 98.5%

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

      6. hypot-def 94.0%

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

      7. unpow2 94.0%

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

      8. unpow2 94.0%

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

      9. +-commutative 94.0%

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

      10. unpow2 94.0%

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

      11. unpow2 94.0%

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

      12. hypot-def 98.5%

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

   8. Simplified 98.5%

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

   if 5.0000000000000001e-4 < (sin.f64 th)

   1. Initial program 91.7%

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

      1. associate-*l/ 91.7%

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

      2. *-commutative 91.7%

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

      3. associate-*l/ 91.6%

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

      4. +-commutative 91.6%

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

      5. unpow2 91.6%

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

      6. unpow2 91.6%

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

      7. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 33.8%

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

      1. add-sqr-sqrt 32.9%

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

      2. sqrt-unprod 48.7%

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

      3. pow2 48.7%

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

   6. Applied egg-rr 48.7%

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

      1. unpow2 48.7%

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

      2. rem-sqrt-square 55.1%

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

   8. Simplified 55.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.02:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.0005:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \end{array} \]

Alternative 7: 46.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-52}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-109}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-52)
   (fabs (sin th))
   (if (<= (sin ky) 2e-109) (* ky (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-52) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-109) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-52)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-109) then
        tmp = ky * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-52) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-109) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-52:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-109:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-52)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-109)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-52)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-109)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-52], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-109], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1e-52

   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/ 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.8%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 29.2%

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

      3. pow2 29.2%

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

      4. *-commutative 29.2%

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

      5. clear-num 29.2%

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

      6. un-div-inv 29.2%

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

   6. Applied egg-rr 29.2%

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

      1. unpow2 29.2%

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

      2. rem-sqrt-square 36.4%

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

      3. associate-/r/ 36.5%

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

      4. *-inverses 36.5%

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

      5. *-lft-identity 36.5%

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

   8. Simplified 36.5%

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

   if -1e-52 < (sin.f64 ky) < 2e-109

   1. Initial program 84.4%

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

      1. *-commutative 84.4%

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

      2. clear-num 84.3%

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

      3. +-commutative 84.3%

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

      4. unpow2 84.3%

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

      5. unpow2 84.3%

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

      6. hypot-udef 99.5%

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

      7. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. hypot-def 84.4%

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

      3. unpow2 84.4%

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

      4. unpow2 84.4%

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

      5. +-commutative 84.4%

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

      6. unpow2 84.4%

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

      7. unpow2 84.4%

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

      8. hypot-def 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in ky around 0 52.9%

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

      1. associate-*r/ 55.6%

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

   10. Simplified 55.6%

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

   if 2e-109 < (sin.f64 ky)

   1. Initial program 99.8%

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

   2. Taylor expanded in kx around 0 61.2%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-52}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-109}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8: 46.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-52}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-109}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-52)
   (fabs (sin th))
   (if (<= (sin ky) 2e-109) (/ ky (/ (sin kx) (sin th))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-52) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-109) {
		tmp = ky / (sin(kx) / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-52)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-109) then
        tmp = ky / (sin(kx) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-52) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-109) {
		tmp = ky / (Math.sin(kx) / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-52:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-109:
		tmp = ky / (math.sin(kx) / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-52)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-109)
		tmp = Float64(ky / Float64(sin(kx) / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-52)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-109)
		tmp = ky / (sin(kx) / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-52], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-109], N[(ky / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1e-52

   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/ 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.8%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 29.2%

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

      3. pow2 29.2%

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

      4. *-commutative 29.2%

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

      5. clear-num 29.2%

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

      6. un-div-inv 29.2%

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

   6. Applied egg-rr 29.2%

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

      1. unpow2 29.2%

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

      2. rem-sqrt-square 36.4%

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

      3. associate-/r/ 36.5%

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

      4. *-inverses 36.5%

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

      5. *-lft-identity 36.5%

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

   8. Simplified 36.5%

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

   if -1e-52 < (sin.f64 ky) < 2e-109

   1. Initial program 84.4%

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

   2. Taylor expanded in ky around 0 52.9%

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

      1. associate-/l* 55.6%

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

      2. associate-/r/ 55.5%

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

   4. Simplified 55.5%

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

      1. associate-*l/ 52.9%

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

      2. associate-/l* 55.6%

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

   6. Applied egg-rr 55.6%

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

   if 2e-109 < (sin.f64 ky)

   1. Initial program 99.8%

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

   2. Taylor expanded in kx around 0 61.2%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-52}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-109}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9: 46.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-52}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-52)
   (fabs (sin th))
   (if (<= (sin ky) 2e-109) (/ (sin th) (/ (sin kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-52) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-109) {
		tmp = sin(th) / (sin(kx) / ky);
	} 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) <= (-1d-52)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-109) then
        tmp = sin(th) / (sin(kx) / ky)
    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) <= -1e-52) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-109) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-52:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-109:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-52)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-109)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-52)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-109)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-52], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-109], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1e-52

   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/ 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.8%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 29.2%

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

      3. pow2 29.2%

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

      4. *-commutative 29.2%

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

      5. clear-num 29.2%

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

      6. un-div-inv 29.2%

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

   6. Applied egg-rr 29.2%

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

      1. unpow2 29.2%

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

      2. rem-sqrt-square 36.4%

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

      3. associate-/r/ 36.5%

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

      4. *-inverses 36.5%

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

      5. *-lft-identity 36.5%

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

   8. Simplified 36.5%

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

   if -1e-52 < (sin.f64 ky) < 2e-109

   1. Initial program 84.4%

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

      1. *-commutative 84.4%

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

      2. clear-num 84.3%

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

      3. +-commutative 84.3%

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

      4. unpow2 84.3%

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

      5. unpow2 84.3%

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

      6. hypot-udef 99.5%

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

      7. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in ky around 0 55.7%

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

   if 2e-109 < (sin.f64 ky)

   1. Initial program 99.8%

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

   2. Taylor expanded in kx around 0 61.2%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-52}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 41.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-106}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -4e-106)
   (fabs (sin th))
   (if (<= (sin ky) 1e-104) (/ (sin th) (/ kx ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -4e-106) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-104) {
		tmp = sin(th) / (kx / ky);
	} 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) <= (-4d-106)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-104) then
        tmp = sin(th) / (kx / ky)
    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) <= -4e-106) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-104) {
		tmp = Math.sin(th) / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -4e-106:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-104:
		tmp = math.sin(th) / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -4e-106)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-104)
		tmp = Float64(sin(th) / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -4e-106)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-104)
		tmp = sin(th) / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -4e-106], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-104], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -3.99999999999999976e-106

   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/ 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.8%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 27.4%

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

      3. pow2 27.4%

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

      4. *-commutative 27.4%

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

      5. clear-num 27.4%

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

      6. un-div-inv 27.4%

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

   6. Applied egg-rr 27.4%

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

      1. unpow2 27.4%

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

      2. rem-sqrt-square 34.9%

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

      3. associate-/r/ 35.0%

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

      4. *-inverses 35.0%

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

      5. *-lft-identity 35.0%

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

   8. Simplified 35.0%

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

   if -3.99999999999999976e-106 < (sin.f64 ky) < 9.99999999999999927e-105

   1. Initial program 82.7%

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

   2. Taylor expanded in ky around 0 56.3%

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

      1. associate-/l* 59.2%

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

      2. associate-/r/ 59.2%

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

   4. Simplified 59.2%

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

   5. Taylor expanded in kx around 0 41.3%

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

      1. *-commutative 41.3%

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

      2. clear-num 41.3%

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

      3. un-div-inv 41.4%

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

   7. Applied egg-rr 41.4%

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

   if 9.99999999999999927e-105 < (sin.f64 ky)

   1. Initial program 99.8%

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

   2. Taylor expanded in kx around 0 61.8%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-106}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 33.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-104) (/ ky (/ kx (sin th))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-104) {
		tmp = ky / (kx / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 1d-104) then
        tmp = ky / (kx / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 1e-104) {
		tmp = ky / (kx / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-104:
		tmp = ky / (kx / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-104)
		tmp = Float64(ky / Float64(kx / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-104)
		tmp = ky / (kx / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-104], N[(ky / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.99999999999999927e-105

   1. Initial program 91.4%

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

   2. Taylor expanded in ky around 0 31.8%

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

      1. associate-/l* 33.2%

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

      2. associate-/r/ 33.2%

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

   4. Simplified 33.2%

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

   5. Taylor expanded in kx around 0 21.0%

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

      1. associate-/l* 22.5%

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

   7. Simplified 22.5%

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

   if 9.99999999999999927e-105 < (sin.f64 ky)

   1. Initial program 99.8%

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

   2. Taylor expanded in kx around 0 61.8%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 33.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-104) (/ (sin th) (/ kx ky)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-104) {
		tmp = sin(th) / (kx / ky);
	} 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) <= 1d-104) then
        tmp = sin(th) / (kx / ky)
    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) <= 1e-104) {
		tmp = Math.sin(th) / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-104:
		tmp = math.sin(th) / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-104)
		tmp = Float64(sin(th) / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-104)
		tmp = sin(th) / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-104], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.99999999999999927e-105

   1. Initial program 91.4%

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

   2. Taylor expanded in ky around 0 31.8%

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

      1. associate-/l* 33.2%

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

      2. associate-/r/ 33.2%

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

   4. Simplified 33.2%

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

   5. Taylor expanded in kx around 0 22.5%

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

      1. *-commutative 22.5%

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

      2. clear-num 22.5%

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

      3. un-div-inv 22.5%

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

   7. Applied egg-rr 22.5%

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

   if 9.99999999999999927e-105 < (sin.f64 ky)

   1. Initial program 99.8%

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

   2. Taylor expanded in kx around 0 61.8%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 33.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -650000 \lor \neg \left(ky \leq 3.8 \cdot 10^{-102}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -650000.0) (not (<= ky 3.8e-102)))
   (sin th)
   (* (sin th) (/ ky kx))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -650000.0) || !(ky <= 3.8e-102)) {
		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 <= (-650000.0d0)) .or. (.not. (ky <= 3.8d-102))) 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 <= -650000.0) || !(ky <= 3.8e-102)) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (ky / kx);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -650000.0) or not (ky <= 3.8e-102):
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (ky / kx)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -650000.0) || !(ky <= 3.8e-102))
		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 <= -650000.0) || ~((ky <= 3.8e-102)))
		tmp = sin(th);
	else
		tmp = sin(th) * (ky / kx);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -650000.0], N[Not[LessEqual[ky, 3.8e-102]], $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 < -6.5e5 or 3.80000000000000026e-102 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 37.9%

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

   if -6.5e5 < ky < 3.80000000000000026e-102

   1. Initial program 86.1%

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

   2. Taylor expanded in ky around 0 50.2%

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

      1. associate-/l* 52.6%

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

      2. associate-/r/ 52.6%

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

   4. Simplified 52.6%

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

   5. Taylor expanded in kx around 0 34.6%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -650000 \lor \neg \left(ky \leq 3.8 \cdot 10^{-102}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \end{array} \]

Alternative 14: 30.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -650000 \lor \neg \left(ky \leq 7.4 \cdot 10^{-159}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -650000.0) (not (<= ky 7.4e-159))) (sin th) (/ ky (/ kx th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -650000.0) || !(ky <= 7.4e-159)) {
		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 <= (-650000.0d0)) .or. (.not. (ky <= 7.4d-159))) 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 <= -650000.0) || !(ky <= 7.4e-159)) {
		tmp = Math.sin(th);
	} else {
		tmp = ky / (kx / th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -650000.0) or not (ky <= 7.4e-159):
		tmp = math.sin(th)
	else:
		tmp = ky / (kx / th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -650000.0) || !(ky <= 7.4e-159))
		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 <= -650000.0) || ~((ky <= 7.4e-159)))
		tmp = sin(th);
	else
		tmp = ky / (kx / th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -650000.0], N[Not[LessEqual[ky, 7.4e-159]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(ky / N[(kx / th), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -6.5e5 or 7.3999999999999998e-159 < ky

   1. Initial program 99.5%

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

   2. Taylor expanded in kx around 0 36.7%

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

   if -6.5e5 < ky < 7.3999999999999998e-159

   1. Initial program 84.6%

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

   2. Taylor expanded in ky around 0 50.0%

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

      1. associate-/l* 52.7%

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

      2. associate-/r/ 52.7%

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

   4. Simplified 52.7%

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

   5. Taylor expanded in kx around 0 38.8%

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

   6. Taylor expanded in th around 0 24.7%

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

      1. associate-/l* 27.5%

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

   8. Simplified 27.5%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -650000 \lor \neg \left(ky \leq 7.4 \cdot 10^{-159}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \end{array} \]

Alternative 15: 21.7% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -550000:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 4.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -550000.0) th (if (<= ky 4.8e-57) (/ ky (/ kx th)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -550000.0) {
		tmp = th;
	} else if (ky <= 4.8e-57) {
		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 <= (-550000.0d0)) then
        tmp = th
    else if (ky <= 4.8d-57) 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 <= -550000.0) {
		tmp = th;
	} else if (ky <= 4.8e-57) {
		tmp = ky / (kx / th);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -550000.0:
		tmp = th
	elif ky <= 4.8e-57:
		tmp = ky / (kx / th)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -550000.0)
		tmp = th;
	elseif (ky <= 4.8e-57)
		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 <= -550000.0)
		tmp = th;
	elseif (ky <= 4.8e-57)
		tmp = ky / (kx / th);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -550000.0], th, If[LessEqual[ky, 4.8e-57], N[(ky / N[(kx / th), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -5.5e5 or 4.80000000000000012e-57 < 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/ 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 36.7%

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

   5. Taylor expanded in th around 0 20.0%

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

   if -5.5e5 < ky < 4.80000000000000012e-57

   1. Initial program 87.4%

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

   2. Taylor expanded in ky around 0 46.8%

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

      1. associate-/l* 48.9%

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

      2. associate-/r/ 48.9%

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

   4. Simplified 48.9%

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

   5. Taylor expanded in kx around 0 32.7%

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

   6. Taylor expanded in th around 0 21.4%

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

      1. associate-/l* 23.6%

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

   8. Simplified 23.6%

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

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -550000:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 4.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 16: 13.7% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 94.5%

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

   1. associate-*l/ 93.8%

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

   2. *-commutative 93.8%

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

   3. associate-*l/ 94.4%

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

   4. +-commutative 94.4%

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

   5. unpow2 94.4%

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

   6. unpow2 94.4%

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

   7. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in kx around 0 26.1%

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

5. Taylor expanded in th around 0 14.0%

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

6. Final simplification 14.0%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023307 
(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)))