Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.6% → 99.7%

Time: 18.7s

Alternatives: 18

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

   1. +-commutative 93.1%

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

   2. unpow2 93.1%

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

   3. unpow2 93.1%

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

   4. hypot-undefine 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 46.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-202}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-86}:\\ \;\;\;\;\left|\frac{ky}{\frac{\sin kx}{\sin th}}\right|\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-78} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-10}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.005)
   (* (sin ky) (fabs (/ (sin th) (sin ky))))
   (if (<= (sin ky) 1e-202)
     (* (sin th) (/ ky (sin kx)))
     (if (<= (sin ky) 5e-86)
       (fabs (/ ky (/ (sin kx) (sin th))))
       (if (or (<= (sin ky) 3e-78) (not (<= (sin ky) 5e-10)))
         (sin th)
         (fabs (* (sin th) (/ (sin ky) (sin kx)))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.005) {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	} else if (sin(ky) <= 1e-202) {
		tmp = sin(th) * (ky / sin(kx));
	} else if (sin(ky) <= 5e-86) {
		tmp = fabs((ky / (sin(kx) / sin(th))));
	} else if ((sin(ky) <= 3e-78) || !(sin(ky) <= 5e-10)) {
		tmp = sin(th);
	} else {
		tmp = fabs((sin(th) * (sin(ky) / sin(kx))));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.005d0)) then
        tmp = sin(ky) * abs((sin(th) / sin(ky)))
    else if (sin(ky) <= 1d-202) then
        tmp = sin(th) * (ky / sin(kx))
    else if (sin(ky) <= 5d-86) then
        tmp = abs((ky / (sin(kx) / sin(th))))
    else if ((sin(ky) <= 3d-78) .or. (.not. (sin(ky) <= 5d-10))) then
        tmp = sin(th)
    else
        tmp = abs((sin(th) * (sin(ky) / sin(kx))))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.005) {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	} else if (Math.sin(ky) <= 1e-202) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else if (Math.sin(ky) <= 5e-86) {
		tmp = Math.abs((ky / (Math.sin(kx) / Math.sin(th))));
	} else if ((Math.sin(ky) <= 3e-78) || !(Math.sin(ky) <= 5e-10)) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.abs((Math.sin(th) * (Math.sin(ky) / Math.sin(kx))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.005:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	elif math.sin(ky) <= 1e-202:
		tmp = math.sin(th) * (ky / math.sin(kx))
	elif math.sin(ky) <= 5e-86:
		tmp = math.fabs((ky / (math.sin(kx) / math.sin(th))))
	elif (math.sin(ky) <= 3e-78) or not (math.sin(ky) <= 5e-10):
		tmp = math.sin(th)
	else:
		tmp = math.fabs((math.sin(th) * (math.sin(ky) / math.sin(kx))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.005)
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	elseif (sin(ky) <= 1e-202)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	elseif (sin(ky) <= 5e-86)
		tmp = abs(Float64(ky / Float64(sin(kx) / sin(th))));
	elseif ((sin(ky) <= 3e-78) || !(sin(ky) <= 5e-10))
		tmp = sin(th);
	else
		tmp = abs(Float64(sin(th) * Float64(sin(ky) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.005)
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	elseif (sin(ky) <= 1e-202)
		tmp = sin(th) * (ky / sin(kx));
	elseif (sin(ky) <= 5e-86)
		tmp = abs((ky / (sin(kx) / sin(th))));
	elseif ((sin(ky) <= 3e-78) || ~((sin(ky) <= 5e-10)))
		tmp = sin(th);
	else
		tmp = abs((sin(th) * (sin(ky) / sin(kx))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.005], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-202], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-86], N[Abs[N[(ky / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[N[Sin[ky], $MachinePrecision], 3e-78], N[Not[LessEqual[N[Sin[ky], $MachinePrecision], 5e-10]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[Abs[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 ky) < -0.0050000000000000001

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.1%

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

      1. add-sqr-sqrt 0.9%

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

      2. sqrt-unprod 35.2%

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

      3. pow2 35.2%

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

   7. Applied egg-rr 35.2%

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

      1. unpow2 35.2%

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

      2. rem-sqrt-square 40.1%

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

   9. Simplified 40.1%

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

   if -0.0050000000000000001 < (sin.f64 ky) < 1e-202

   1. Initial program 85.0%

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

      1. unpow2 85.0%

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

      2. sqr-neg 85.0%

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

      3. sin-neg 85.0%

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

      4. sin-neg 85.0%

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

      5. unpow2 85.0%

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

      6. associate-*l/ 84.2%

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

      7. associate-/l* 84.9%

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

      8. unpow2 84.9%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 47.1%

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

      1. *-commutative 47.1%

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

      2. associate-/l* 47.6%

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

   7. Simplified 47.6%

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

   if 1e-202 < (sin.f64 ky) < 4.9999999999999999e-86

   1. Initial program 90.0%

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

      1. unpow2 90.0%

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

      2. sqr-neg 90.0%

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

      3. sin-neg 90.0%

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

      4. sin-neg 90.0%

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

      5. unpow2 90.0%

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

      6. associate-*l/ 86.1%

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

      7. associate-/l* 89.8%

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

      8. unpow2 89.8%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in ky around 0 60.3%

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

      1. associate-/l* 60.0%

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

   7. Simplified 60.0%

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

      1. add-sqr-sqrt 42.3%

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

      2. sqrt-unprod 36.1%

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

      3. pow2 36.1%

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

   9. Applied egg-rr 36.1%

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

       1. unpow2 36.1%

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

       2. rem-sqrt-square 54.5%

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

       3. associate-*r/ 54.9%

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

       4. *-rgt-identity 54.9%

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

       5. times-frac 54.7%

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

       6. /-rgt-identity 54.7%

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

       7. associate-/r/ 54.6%

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

   11. Simplified 54.6%

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

   if 4.9999999999999999e-86 < (sin.f64 ky) < 2.99999999999999988e-78 or 5.00000000000000031e-10 < (sin.f64 ky)

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 52.9%

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

   if 2.99999999999999988e-78 < (sin.f64 ky) < 5.00000000000000031e-10

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 35.1%

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

      1. add-sqr-sqrt 18.0%

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

      2. sqrt-unprod 25.3%

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

      3. pow2 25.3%

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

   7. Applied egg-rr 25.3%

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

      1. unpow2 25.3%

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

      2. rem-sqrt-square 35.4%

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

      3. *-commutative 35.4%

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

      4. associate-*l/ 35.4%

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

      5. associate-/l* 35.6%

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

   9. Simplified 35.6%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-202}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-86}:\\ \;\;\;\;\left|\frac{ky}{\frac{\sin kx}{\sin th}}\right|\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-78} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-10}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left|\frac{ky}{\frac{\sin kx}{\sin th}}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-148}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 1.5 \cdot 10^{-121}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-102}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.01)
   (fabs (/ ky (/ (sin kx) (sin th))))
   (if (<= (sin kx) 1e-148)
     (sin th)
     (if (<= (sin kx) 1.5e-121)
       (* (sin ky) (/ (sin th) kx))
       (if (<= (sin kx) 1e-102)
         (/ (* (sin ky) (sin th)) (sin ky))
         (/ (sin th) (/ (sin kx) (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.01) {
		tmp = fabs((ky / (sin(kx) / sin(th))));
	} else if (sin(kx) <= 1e-148) {
		tmp = sin(th);
	} else if (sin(kx) <= 1.5e-121) {
		tmp = sin(ky) * (sin(th) / kx);
	} else if (sin(kx) <= 1e-102) {
		tmp = (sin(ky) * sin(th)) / sin(ky);
	} else {
		tmp = sin(th) / (sin(kx) / sin(ky));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.01d0)) then
        tmp = abs((ky / (sin(kx) / sin(th))))
    else if (sin(kx) <= 1d-148) then
        tmp = sin(th)
    else if (sin(kx) <= 1.5d-121) then
        tmp = sin(ky) * (sin(th) / kx)
    else if (sin(kx) <= 1d-102) then
        tmp = (sin(ky) * sin(th)) / sin(ky)
    else
        tmp = sin(th) / (sin(kx) / sin(ky))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.01) {
		tmp = Math.abs((ky / (Math.sin(kx) / Math.sin(th))));
	} else if (Math.sin(kx) <= 1e-148) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 1.5e-121) {
		tmp = Math.sin(ky) * (Math.sin(th) / kx);
	} else if (Math.sin(kx) <= 1e-102) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sin(ky);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.01:
		tmp = math.fabs((ky / (math.sin(kx) / math.sin(th))))
	elif math.sin(kx) <= 1e-148:
		tmp = math.sin(th)
	elif math.sin(kx) <= 1.5e-121:
		tmp = math.sin(ky) * (math.sin(th) / kx)
	elif math.sin(kx) <= 1e-102:
		tmp = (math.sin(ky) * math.sin(th)) / math.sin(ky)
	else:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.01)
		tmp = abs(Float64(ky / Float64(sin(kx) / sin(th))));
	elseif (sin(kx) <= 1e-148)
		tmp = sin(th);
	elseif (sin(kx) <= 1.5e-121)
		tmp = Float64(sin(ky) * Float64(sin(th) / kx));
	elseif (sin(kx) <= 1e-102)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sin(ky));
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.01)
		tmp = abs((ky / (sin(kx) / sin(th))));
	elseif (sin(kx) <= 1e-148)
		tmp = sin(th);
	elseif (sin(kx) <= 1.5e-121)
		tmp = sin(ky) * (sin(th) / kx);
	elseif (sin(kx) <= 1e-102)
		tmp = (sin(ky) * sin(th)) / sin(ky);
	else
		tmp = sin(th) / (sin(kx) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.01], N[Abs[N[(ky / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-148], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1.5e-121], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-102], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 19.2%

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

      1. associate-/l* 19.2%

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

   7. Simplified 19.2%

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

      1. add-sqr-sqrt 17.3%

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

      2. sqrt-unprod 26.2%

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

      3. pow2 26.2%

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

   9. Applied egg-rr 26.2%

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

       1. unpow2 26.2%

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

       2. rem-sqrt-square 45.3%

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

       3. associate-*r/ 45.3%

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

       4. *-rgt-identity 45.3%

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

       5. times-frac 45.3%

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

       6. /-rgt-identity 45.3%

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

       7. associate-/r/ 45.3%

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

   11. Simplified 45.3%

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

   if -0.0100000000000000002 < (sin.f64 kx) < 9.99999999999999936e-149

   1. Initial program 82.3%

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

      1. unpow2 82.3%

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

      2. sqr-neg 82.3%

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

      3. sin-neg 82.3%

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

      4. sin-neg 82.3%

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

      5. unpow2 82.3%

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

      6. associate-*l/ 81.0%

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

      7. associate-/l* 82.2%

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

      8. unpow2 82.2%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 35.1%

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

   if 9.99999999999999936e-149 < (sin.f64 kx) < 1.5e-121

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 91.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 62.3%

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

   6. Taylor expanded in kx around 0 62.3%

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

   if 1.5e-121 < (sin.f64 kx) < 9.99999999999999933e-103

   1. Initial program 100.0%

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

      1. unpow2 100.0%

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

      2. sqr-neg 100.0%

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

      3. sin-neg 100.0%

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

      4. sin-neg 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/l* 100.0%

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

      8. unpow2 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in kx around 0 34.4%

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

   if 9.99999999999999933e-103 < (sin.f64 kx)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.5%

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

      2. hypot-undefine 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-*l/ 99.6%

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

      7. *-commutative 99.6%

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

      8. clear-num 99.5%

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

      9. un-div-inv 99.5%

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

      10. +-commutative 99.5%

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

      11. unpow2 99.5%

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

      12. unpow2 99.5%

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

      13. hypot-undefine 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ky around 0 60.2%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left|\frac{ky}{\frac{\sin kx}{\sin th}}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-148}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 1.5 \cdot 10^{-121}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-102}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-148}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 1.5 \cdot 10^{-121}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-102}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.01)
   (fabs (* (sin th) (/ (sin ky) (sin kx))))
   (if (<= (sin kx) 1e-148)
     (sin th)
     (if (<= (sin kx) 1.5e-121)
       (* (sin ky) (/ (sin th) kx))
       (if (<= (sin kx) 1e-102)
         (/ (* (sin ky) (sin th)) (sin ky))
         (/ (sin th) (/ (sin kx) (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.01) {
		tmp = fabs((sin(th) * (sin(ky) / sin(kx))));
	} else if (sin(kx) <= 1e-148) {
		tmp = sin(th);
	} else if (sin(kx) <= 1.5e-121) {
		tmp = sin(ky) * (sin(th) / kx);
	} else if (sin(kx) <= 1e-102) {
		tmp = (sin(ky) * sin(th)) / sin(ky);
	} else {
		tmp = sin(th) / (sin(kx) / sin(ky));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.01d0)) then
        tmp = abs((sin(th) * (sin(ky) / sin(kx))))
    else if (sin(kx) <= 1d-148) then
        tmp = sin(th)
    else if (sin(kx) <= 1.5d-121) then
        tmp = sin(ky) * (sin(th) / kx)
    else if (sin(kx) <= 1d-102) then
        tmp = (sin(ky) * sin(th)) / sin(ky)
    else
        tmp = sin(th) / (sin(kx) / sin(ky))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.01) {
		tmp = Math.abs((Math.sin(th) * (Math.sin(ky) / Math.sin(kx))));
	} else if (Math.sin(kx) <= 1e-148) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 1.5e-121) {
		tmp = Math.sin(ky) * (Math.sin(th) / kx);
	} else if (Math.sin(kx) <= 1e-102) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sin(ky);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.01:
		tmp = math.fabs((math.sin(th) * (math.sin(ky) / math.sin(kx))))
	elif math.sin(kx) <= 1e-148:
		tmp = math.sin(th)
	elif math.sin(kx) <= 1.5e-121:
		tmp = math.sin(ky) * (math.sin(th) / kx)
	elif math.sin(kx) <= 1e-102:
		tmp = (math.sin(ky) * math.sin(th)) / math.sin(ky)
	else:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.01)
		tmp = abs(Float64(sin(th) * Float64(sin(ky) / sin(kx))));
	elseif (sin(kx) <= 1e-148)
		tmp = sin(th);
	elseif (sin(kx) <= 1.5e-121)
		tmp = Float64(sin(ky) * Float64(sin(th) / kx));
	elseif (sin(kx) <= 1e-102)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sin(ky));
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.01)
		tmp = abs((sin(th) * (sin(ky) / sin(kx))));
	elseif (sin(kx) <= 1e-148)
		tmp = sin(th);
	elseif (sin(kx) <= 1.5e-121)
		tmp = sin(ky) * (sin(th) / kx);
	elseif (sin(kx) <= 1e-102)
		tmp = (sin(ky) * sin(th)) / sin(ky);
	else
		tmp = sin(th) / (sin(kx) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.01], N[Abs[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-148], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1.5e-121], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-102], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 18.7%

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

      1. add-sqr-sqrt 17.1%

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

      2. sqrt-unprod 29.4%

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

      3. pow2 29.4%

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

   7. Applied egg-rr 29.4%

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

      1. unpow2 29.4%

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

      2. rem-sqrt-square 48.6%

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

      3. *-commutative 48.6%

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

      4. associate-*l/ 48.6%

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

      5. associate-/l* 48.6%

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

   9. Simplified 48.6%

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

   if -0.0100000000000000002 < (sin.f64 kx) < 9.99999999999999936e-149

   1. Initial program 82.3%

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

      1. unpow2 82.3%

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

      2. sqr-neg 82.3%

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

      3. sin-neg 82.3%

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

      4. sin-neg 82.3%

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

      5. unpow2 82.3%

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

      6. associate-*l/ 81.0%

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

      7. associate-/l* 82.2%

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

      8. unpow2 82.2%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 35.1%

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

   if 9.99999999999999936e-149 < (sin.f64 kx) < 1.5e-121

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 91.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 62.3%

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

   6. Taylor expanded in kx around 0 62.3%

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

   if 1.5e-121 < (sin.f64 kx) < 9.99999999999999933e-103

   1. Initial program 100.0%

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

      1. unpow2 100.0%

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

      2. sqr-neg 100.0%

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

      3. sin-neg 100.0%

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

      4. sin-neg 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/l* 100.0%

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

      8. unpow2 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in kx around 0 34.4%

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

   if 9.99999999999999933e-103 < (sin.f64 kx)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.5%

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

      2. hypot-undefine 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-*l/ 99.6%

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

      7. *-commutative 99.6%

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

      8. clear-num 99.5%

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

      9. un-div-inv 99.5%

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

      10. +-commutative 99.5%

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

      11. unpow2 99.5%

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

      12. unpow2 99.5%

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

      13. hypot-undefine 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ky around 0 60.2%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-148}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 1.5 \cdot 10^{-121}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-102}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-148}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 1.5 \cdot 10^{-121}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-102}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.01)
   (* (sin ky) (fabs (/ (sin th) (sin kx))))
   (if (<= (sin kx) 1e-148)
     (sin th)
     (if (<= (sin kx) 1.5e-121)
       (* (sin ky) (/ (sin th) kx))
       (if (<= (sin kx) 1e-102)
         (/ (* (sin ky) (sin th)) (sin ky))
         (/ (sin th) (/ (sin kx) (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.01) {
		tmp = sin(ky) * fabs((sin(th) / sin(kx)));
	} else if (sin(kx) <= 1e-148) {
		tmp = sin(th);
	} else if (sin(kx) <= 1.5e-121) {
		tmp = sin(ky) * (sin(th) / kx);
	} else if (sin(kx) <= 1e-102) {
		tmp = (sin(ky) * sin(th)) / sin(ky);
	} else {
		tmp = sin(th) / (sin(kx) / sin(ky));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.01d0)) then
        tmp = sin(ky) * abs((sin(th) / sin(kx)))
    else if (sin(kx) <= 1d-148) then
        tmp = sin(th)
    else if (sin(kx) <= 1.5d-121) then
        tmp = sin(ky) * (sin(th) / kx)
    else if (sin(kx) <= 1d-102) then
        tmp = (sin(ky) * sin(th)) / sin(ky)
    else
        tmp = sin(th) / (sin(kx) / sin(ky))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.01) {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(kx)));
	} else if (Math.sin(kx) <= 1e-148) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 1.5e-121) {
		tmp = Math.sin(ky) * (Math.sin(th) / kx);
	} else if (Math.sin(kx) <= 1e-102) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sin(ky);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.01:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(kx)))
	elif math.sin(kx) <= 1e-148:
		tmp = math.sin(th)
	elif math.sin(kx) <= 1.5e-121:
		tmp = math.sin(ky) * (math.sin(th) / kx)
	elif math.sin(kx) <= 1e-102:
		tmp = (math.sin(ky) * math.sin(th)) / math.sin(ky)
	else:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.01)
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= 1e-148)
		tmp = sin(th);
	elseif (sin(kx) <= 1.5e-121)
		tmp = Float64(sin(ky) * Float64(sin(th) / kx));
	elseif (sin(kx) <= 1e-102)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sin(ky));
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.01)
		tmp = sin(ky) * abs((sin(th) / sin(kx)));
	elseif (sin(kx) <= 1e-148)
		tmp = sin(th);
	elseif (sin(kx) <= 1.5e-121)
		tmp = sin(ky) * (sin(th) / kx);
	elseif (sin(kx) <= 1e-102)
		tmp = (sin(ky) * sin(th)) / sin(ky);
	else
		tmp = sin(th) / (sin(kx) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.01], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-148], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1.5e-121], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-102], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 18.7%

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

      1. add-sqr-sqrt 13.7%

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

      2. sqrt-unprod 39.9%

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

      3. pow2 39.9%

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

   7. Applied egg-rr 39.9%

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

      1. unpow2 39.9%

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

      2. rem-sqrt-square 39.9%

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

   9. Simplified 39.9%

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

   if -0.0100000000000000002 < (sin.f64 kx) < 9.99999999999999936e-149

   1. Initial program 82.3%

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

      1. unpow2 82.3%

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

      2. sqr-neg 82.3%

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

      3. sin-neg 82.3%

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

      4. sin-neg 82.3%

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

      5. unpow2 82.3%

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

      6. associate-*l/ 81.0%

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

      7. associate-/l* 82.2%

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

      8. unpow2 82.2%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 35.1%

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

   if 9.99999999999999936e-149 < (sin.f64 kx) < 1.5e-121

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 91.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 62.3%

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

   6. Taylor expanded in kx around 0 62.3%

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

   if 1.5e-121 < (sin.f64 kx) < 9.99999999999999933e-103

   1. Initial program 100.0%

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

      1. unpow2 100.0%

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

      2. sqr-neg 100.0%

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

      3. sin-neg 100.0%

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

      4. sin-neg 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/l* 100.0%

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

      8. unpow2 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in kx around 0 34.4%

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

   if 9.99999999999999933e-103 < (sin.f64 kx)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.5%

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

      2. hypot-undefine 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-*l/ 99.6%

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

      7. *-commutative 99.6%

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

      8. clear-num 99.5%

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

      9. un-div-inv 99.5%

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

      10. +-commutative 99.5%

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

      11. unpow2 99.5%

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

      12. unpow2 99.5%

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

      13. hypot-undefine 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ky around 0 60.2%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-148}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 1.5 \cdot 10^{-121}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-102}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.02:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.02)
   (sin th)
   (if (<= (sin th) 2e-8)
     (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
     (* (sin ky) (fabs (/ (sin th) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.02) {
		tmp = sin(th);
	} else if (sin(th) <= 2e-8) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.02) {
		tmp = Math.sin(th);
	} else if (Math.sin(th) <= 2e-8) {
		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(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.02:
		tmp = math.sin(th)
	elif math.sin(th) <= 2e-8:
		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(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.02)
		tmp = sin(th);
	elseif (sin(th) <= 2e-8)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.02)
		tmp = sin(th);
	elseif (sin(th) <= 2e-8)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	else
		tmp = sin(ky) * abs((sin(th) / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.02], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 2e-8], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 90.1%

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

      1. unpow2 90.1%

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

      2. sqr-neg 90.1%

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

      3. sin-neg 90.1%

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

      4. sin-neg 90.1%

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

      5. unpow2 90.1%

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

      6. associate-*l/ 90.0%

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

      7. associate-/l* 90.0%

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

      8. unpow2 90.0%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in kx around 0 17.0%

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

   if -0.0200000000000000004 < (sin.f64 th) < 2e-8

   1. Initial program 93.4%

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

      1. unpow2 93.4%

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

      2. sqr-neg 93.4%

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

      3. sin-neg 93.4%

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

      4. sin-neg 93.4%

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

      5. unpow2 93.4%

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

      6. associate-*l/ 92.0%

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

      7. associate-/l* 93.4%

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

      8. unpow2 93.4%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 92.8%

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

   6. Applied egg-rr 92.8%

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

   7. Taylor expanded in th around 0 92.6%

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

      1. associate-/l* 99.6%

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

      2. *-commutative 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 2e-8 < (sin.f64 th)

   1. Initial program 94.7%

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

      1. unpow2 94.7%

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

      2. sqr-neg 94.7%

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

      3. sin-neg 94.7%

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

      4. sin-neg 94.7%

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

      5. unpow2 94.7%

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

      6. associate-*l/ 94.4%

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

      7. associate-/l* 94.6%

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

      8. unpow2 94.6%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ky around 0 28.7%

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

      1. add-sqr-sqrt 27.8%

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

      2. sqrt-unprod 46.4%

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

      3. pow2 46.4%

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

   7. Applied egg-rr 46.4%

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

      1. unpow2 46.4%

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

      2. rem-sqrt-square 49.0%

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

   9. Simplified 49.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.02:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-165}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-135} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-86}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.005)
   (fabs (sin th))
   (if (<= (sin ky) 5e-165)
     (* ky (/ (sin th) kx))
     (if (or (<= (sin ky) 4e-135) (not (<= (sin ky) 5e-86)))
       (sin th)
       (/ (* ky (sin th)) kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.005) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-165) {
		tmp = ky * (sin(th) / kx);
	} else if ((sin(ky) <= 4e-135) || !(sin(ky) <= 5e-86)) {
		tmp = sin(th);
	} else {
		tmp = (ky * sin(th)) / 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 (sin(ky) <= (-0.005d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-165) then
        tmp = ky * (sin(th) / kx)
    else if ((sin(ky) <= 4d-135) .or. (.not. (sin(ky) <= 5d-86))) then
        tmp = sin(th)
    else
        tmp = (ky * sin(th)) / kx
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.005) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-165) {
		tmp = ky * (Math.sin(th) / kx);
	} else if ((Math.sin(ky) <= 4e-135) || !(Math.sin(ky) <= 5e-86)) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky * Math.sin(th)) / kx;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.005:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-165:
		tmp = ky * (math.sin(th) / kx)
	elif (math.sin(ky) <= 4e-135) or not (math.sin(ky) <= 5e-86):
		tmp = math.sin(th)
	else:
		tmp = (ky * math.sin(th)) / kx
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-165)
		tmp = Float64(ky * Float64(sin(th) / kx));
	elseif ((sin(ky) <= 4e-135) || !(sin(ky) <= 5e-86))
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky * sin(th)) / kx);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-165)
		tmp = ky * (sin(th) / kx);
	elseif ((sin(ky) <= 4e-135) || ~((sin(ky) <= 5e-86)))
		tmp = sin(th);
	else
		tmp = (ky * sin(th)) / kx;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.005], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-165], N[(ky * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[Sin[ky], $MachinePrecision], 4e-135], N[Not[LessEqual[N[Sin[ky], $MachinePrecision], 5e-86]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.0050000000000000001

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.1%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 32.4%

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

      3. pow2 32.4%

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

      4. clear-num 32.3%

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

      5. un-div-inv 32.4%

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

   7. Applied egg-rr 32.4%

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

      1. unpow2 32.4%

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

      2. rem-sqrt-square 34.3%

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

      3. associate-/r/ 34.4%

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

      4. *-inverses 34.4%

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

      5. *-lft-identity 34.4%

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

   9. Simplified 34.4%

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

   if -0.0050000000000000001 < (sin.f64 ky) < 4.99999999999999981e-165

   1. Initial program 84.6%

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

      1. unpow2 84.6%

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

      2. sqr-neg 84.6%

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

      3. sin-neg 84.6%

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

      4. sin-neg 84.6%

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

      5. unpow2 84.6%

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

      6. associate-*l/ 83.8%

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

      7. associate-/l* 84.5%

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

      8. unpow2 84.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 49.4%

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

      1. associate-/l* 49.7%

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

   7. Simplified 49.7%

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

   8. Taylor expanded in kx around 0 31.1%

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

      1. associate-/l* 31.4%

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

   10. Simplified 31.4%

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

   if 4.99999999999999981e-165 < (sin.f64 ky) < 4.0000000000000002e-135 or 4.9999999999999999e-86 < (sin.f64 ky)

   1. Initial program 99.1%

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

      1. unpow2 99.1%

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

      2. sqr-neg 99.1%

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

      3. sin-neg 99.1%

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

      4. sin-neg 99.1%

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

      5. unpow2 99.1%

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

      6. associate-*l/ 99.0%

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

      7. associate-/l* 99.1%

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

      8. unpow2 99.1%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 49.0%

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

   if 4.0000000000000002e-135 < (sin.f64 ky) < 4.9999999999999999e-86

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 88.2%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 51.9%

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

      1. associate-/l* 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in kx around 0 36.9%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-165}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-135} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-86}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left|\frac{ky}{\frac{\sin kx}{\sin th}}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-148}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.01)
   (fabs (/ ky (/ (sin kx) (sin th))))
   (if (<= (sin kx) 1e-148) (sin th) (* (sin ky) (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.01) {
		tmp = fabs((ky / (sin(kx) / sin(th))));
	} else if (sin(kx) <= 1e-148) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.01d0)) then
        tmp = abs((ky / (sin(kx) / sin(th))))
    else if (sin(kx) <= 1d-148) then
        tmp = sin(th)
    else
        tmp = sin(ky) * (sin(th) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.01) {
		tmp = Math.abs((ky / (Math.sin(kx) / Math.sin(th))));
	} else if (Math.sin(kx) <= 1e-148) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.01:
		tmp = math.fabs((ky / (math.sin(kx) / math.sin(th))))
	elif math.sin(kx) <= 1e-148:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.01)
		tmp = abs(Float64(ky / Float64(sin(kx) / sin(th))));
	elseif (sin(kx) <= 1e-148)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.01)
		tmp = abs((ky / (sin(kx) / sin(th))));
	elseif (sin(kx) <= 1e-148)
		tmp = sin(th);
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.01], N[Abs[N[(ky / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-148], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 19.2%

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

      1. associate-/l* 19.2%

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

   7. Simplified 19.2%

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

      1. add-sqr-sqrt 17.3%

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

      2. sqrt-unprod 26.2%

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

      3. pow2 26.2%

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

   9. Applied egg-rr 26.2%

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

       1. unpow2 26.2%

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

       2. rem-sqrt-square 45.3%

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

       3. associate-*r/ 45.3%

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

       4. *-rgt-identity 45.3%

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

       5. times-frac 45.3%

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

       6. /-rgt-identity 45.3%

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

       7. associate-/r/ 45.3%

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

   11. Simplified 45.3%

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

   if -0.0100000000000000002 < (sin.f64 kx) < 9.99999999999999936e-149

   1. Initial program 82.3%

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

      1. unpow2 82.3%

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

      2. sqr-neg 82.3%

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

      3. sin-neg 82.3%

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

      4. sin-neg 82.3%

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

      5. unpow2 82.3%

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

      6. associate-*l/ 81.0%

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

      7. associate-/l* 82.2%

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

      8. unpow2 82.2%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 35.1%

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

   if 9.99999999999999936e-149 < (sin.f64 kx)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.0%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 58.7%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left|\frac{ky}{\frac{\sin kx}{\sin th}}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-148}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left|\frac{ky}{\frac{\sin kx}{\sin th}}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-148}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.01)
   (fabs (/ ky (/ (sin kx) (sin th))))
   (if (<= (sin kx) 1e-148) (sin th) (/ (sin th) (/ (sin kx) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.01) {
		tmp = fabs((ky / (sin(kx) / sin(th))));
	} else if (sin(kx) <= 1e-148) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(kx) / sin(ky));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.01d0)) then
        tmp = abs((ky / (sin(kx) / sin(th))))
    else if (sin(kx) <= 1d-148) then
        tmp = sin(th)
    else
        tmp = sin(th) / (sin(kx) / sin(ky))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.01) {
		tmp = Math.abs((ky / (Math.sin(kx) / Math.sin(th))));
	} else if (Math.sin(kx) <= 1e-148) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.01:
		tmp = math.fabs((ky / (math.sin(kx) / math.sin(th))))
	elif math.sin(kx) <= 1e-148:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.01)
		tmp = abs(Float64(ky / Float64(sin(kx) / sin(th))));
	elseif (sin(kx) <= 1e-148)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.01)
		tmp = abs((ky / (sin(kx) / sin(th))));
	elseif (sin(kx) <= 1e-148)
		tmp = sin(th);
	else
		tmp = sin(th) / (sin(kx) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.01], N[Abs[N[(ky / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-148], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 19.2%

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

      1. associate-/l* 19.2%

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

   7. Simplified 19.2%

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

      1. add-sqr-sqrt 17.3%

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

      2. sqrt-unprod 26.2%

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

      3. pow2 26.2%

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

   9. Applied egg-rr 26.2%

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

       1. unpow2 26.2%

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

       2. rem-sqrt-square 45.3%

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

       3. associate-*r/ 45.3%

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

       4. *-rgt-identity 45.3%

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

       5. times-frac 45.3%

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

       6. /-rgt-identity 45.3%

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

       7. associate-/r/ 45.3%

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

   11. Simplified 45.3%

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

   if -0.0100000000000000002 < (sin.f64 kx) < 9.99999999999999936e-149

   1. Initial program 82.3%

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

      1. unpow2 82.3%

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

      2. sqr-neg 82.3%

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

      3. sin-neg 82.3%

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

      4. sin-neg 82.3%

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

      5. unpow2 82.3%

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

      6. associate-*l/ 81.0%

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

      7. associate-/l* 82.2%

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

      8. unpow2 82.2%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 35.1%

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

   if 9.99999999999999936e-149 < (sin.f64 kx)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.0%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.0%

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

      2. hypot-undefine 99.0%

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

      3. unpow2 99.0%

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

      4. unpow2 99.0%

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

      5. +-commutative 99.0%

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

      6. associate-*l/ 99.6%

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

      7. *-commutative 99.6%

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

      8. clear-num 99.5%

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

      9. un-div-inv 99.5%

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

      10. +-commutative 99.5%

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

      11. unpow2 99.5%

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

      12. unpow2 99.5%

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

      13. hypot-undefine 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ky around 0 58.7%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left|\frac{ky}{\frac{\sin kx}{\sin th}}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-148}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

   1. unpow2 93.1%

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

   2. sqr-neg 93.1%

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

   3. sin-neg 93.1%

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

   4. sin-neg 93.1%

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

   5. unpow2 93.1%

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

   6. associate-*l/ 92.3%

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

   7. associate-/l* 93.0%

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

   8. unpow2 93.0%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 11: 47.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-86}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.005)
   (fabs (sin th))
   (if (<= (sin ky) 5e-86) (* ky (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.005) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-86) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.005d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-86) then
        tmp = ky * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.005) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-86) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.005:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-86:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-86)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-86)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.005], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-86], 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) < -0.0050000000000000001

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.1%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 32.4%

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

      3. pow2 32.4%

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

      4. clear-num 32.3%

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

      5. un-div-inv 32.4%

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

   7. Applied egg-rr 32.4%

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

      1. unpow2 32.4%

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

      2. rem-sqrt-square 34.3%

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

      3. associate-/r/ 34.4%

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

      4. *-inverses 34.4%

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

      5. *-lft-identity 34.4%

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

   9. Simplified 34.4%

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

   if -0.0050000000000000001 < (sin.f64 ky) < 4.9999999999999999e-86

   1. Initial program 86.0%

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

      1. unpow2 86.0%

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

      2. sqr-neg 86.0%

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

      3. sin-neg 86.0%

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

      4. sin-neg 86.0%

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

      5. unpow2 86.0%

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

      6. associate-*l/ 84.6%

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

      7. associate-/l* 85.9%

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

      8. unpow2 85.9%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 49.7%

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

      1. associate-/l* 49.9%

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

   7. Simplified 49.9%

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

   if 4.9999999999999999e-86 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 49.8%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-86}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-86}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.005)
   (fabs (sin th))
   (if (<= (sin ky) 5e-86) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.005) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-86) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.005d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-86) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.005) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-86) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.005:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-86:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-86)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-86)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.005], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-86], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0050000000000000001

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.1%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 32.4%

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

      3. pow2 32.4%

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

      4. clear-num 32.3%

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

      5. un-div-inv 32.4%

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

   7. Applied egg-rr 32.4%

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

      1. unpow2 32.4%

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

      2. rem-sqrt-square 34.3%

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

      3. associate-/r/ 34.4%

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

      4. *-inverses 34.4%

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

      5. *-lft-identity 34.4%

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

   9. Simplified 34.4%

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

   if -0.0050000000000000001 < (sin.f64 ky) < 4.9999999999999999e-86

   1. Initial program 86.0%

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

      1. unpow2 86.0%

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

      2. sqr-neg 86.0%

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

      3. sin-neg 86.0%

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

      4. sin-neg 86.0%

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

      5. unpow2 86.0%

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

      6. associate-*l/ 84.6%

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

      7. associate-/l* 85.9%

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

      8. unpow2 85.9%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 49.7%

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

      1. *-commutative 49.7%

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

      2. associate-/l* 50.0%

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

   7. Simplified 50.0%

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

   if 4.9999999999999999e-86 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 49.8%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-86}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left|\frac{ky}{\frac{\sin kx}{\sin th}}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.01)
   (fabs (/ ky (/ (sin kx) (sin th))))
   (if (<= (sin kx) 5e-141) (sin th) (* (sin th) (/ ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.01) {
		tmp = fabs((ky / (sin(kx) / sin(th))));
	} else if (sin(kx) <= 5e-141) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (ky / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.01d0)) then
        tmp = abs((ky / (sin(kx) / sin(th))))
    else if (sin(kx) <= 5d-141) then
        tmp = sin(th)
    else
        tmp = sin(th) * (ky / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.01) {
		tmp = Math.abs((ky / (Math.sin(kx) / Math.sin(th))));
	} else if (Math.sin(kx) <= 5e-141) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.01:
		tmp = math.fabs((ky / (math.sin(kx) / math.sin(th))))
	elif math.sin(kx) <= 5e-141:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.01)
		tmp = abs(Float64(ky / Float64(sin(kx) / sin(th))));
	elseif (sin(kx) <= 5e-141)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.01)
		tmp = abs((ky / (sin(kx) / sin(th))));
	elseif (sin(kx) <= 5e-141)
		tmp = sin(th);
	else
		tmp = sin(th) * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.01], N[Abs[N[(ky / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-141], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 19.2%

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

      1. associate-/l* 19.2%

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

   7. Simplified 19.2%

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

      1. add-sqr-sqrt 17.3%

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

      2. sqrt-unprod 26.2%

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

      3. pow2 26.2%

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

   9. Applied egg-rr 26.2%

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

       1. unpow2 26.2%

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

       2. rem-sqrt-square 45.3%

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

       3. associate-*r/ 45.3%

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

       4. *-rgt-identity 45.3%

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

       5. times-frac 45.3%

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

       6. /-rgt-identity 45.3%

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

       7. associate-/r/ 45.3%

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

   11. Simplified 45.3%

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

   if -0.0100000000000000002 < (sin.f64 kx) < 4.9999999999999999e-141

   1. Initial program 82.7%

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

      1. unpow2 82.7%

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

      2. sqr-neg 82.7%

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

      3. sin-neg 82.7%

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

      4. sin-neg 82.7%

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

      5. unpow2 82.7%

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

      6. associate-*l/ 81.4%

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

      7. associate-/l* 82.5%

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

      8. unpow2 82.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 35.4%

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

   if 4.9999999999999999e-141 < (sin.f64 kx)

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.0%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 52.7%

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

      1. *-commutative 52.7%

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

      2. associate-/l* 53.3%

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

   7. Simplified 53.3%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.01:\\ \;\;\;\;\left|\frac{ky}{\frac{\sin kx}{\sin th}}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;th \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{t\_1} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= th 3e-6) (* (/ (sin ky) t_1) th) (/ (* ky (sin th)) t_1))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (th <= 3e-6) {
		tmp = (sin(ky) / t_1) * th;
	} else {
		tmp = (ky * sin(th)) / t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (th <= 3e-6) {
		tmp = (Math.sin(ky) / t_1) * th;
	} else {
		tmp = (ky * Math.sin(th)) / t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if th <= 3e-6:
		tmp = (math.sin(ky) / t_1) * th
	else:
		tmp = (ky * math.sin(th)) / t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (th <= 3e-6)
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	else
		tmp = Float64(Float64(ky * sin(th)) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (th <= 3e-6)
		tmp = (sin(ky) / t_1) * th;
	else
		tmp = (ky * sin(th)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[th, 3e-6], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 3.0000000000000001e-6

   1. Initial program 94.7%

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

      1. unpow2 94.7%

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

      2. sqr-neg 94.7%

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

      3. sin-neg 94.7%

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

      4. sin-neg 94.7%

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

      5. unpow2 94.7%

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

      6. associate-*l/ 93.7%

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

      7. associate-/l* 94.6%

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

      8. unpow2 94.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 95.3%

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

   6. Applied egg-rr 95.3%

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

   7. Taylor expanded in th around 0 58.6%

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

      1. associate-/l* 63.0%

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

      2. *-commutative 63.0%

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

   9. Applied egg-rr 63.0%

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

   if 3.0000000000000001e-6 < th

   1. Initial program 89.7%

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

      1. unpow2 89.7%

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

      2. sqr-neg 89.7%

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

      3. sin-neg 89.7%

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

      4. sin-neg 89.7%

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

      5. unpow2 89.7%

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

      6. associate-*l/ 89.5%

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

      7. associate-/l* 89.5%

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

      8. unpow2 89.5%

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

   3. Simplified 99.4%

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

      1. associate-*r/ 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in ky around 0 56.0%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.26 \cdot 10^{-138}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 1.05:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.26e-138)
   (sin th)
   (if (<= kx 1.05) (* ky (/ (sin th) kx)) (* ky (/ th (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.26e-138) {
		tmp = sin(th);
	} else if (kx <= 1.05) {
		tmp = ky * (sin(th) / kx);
	} else {
		tmp = ky * (th / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 1.26d-138) then
        tmp = sin(th)
    else if (kx <= 1.05d0) then
        tmp = ky * (sin(th) / kx)
    else
        tmp = ky * (th / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.26e-138) {
		tmp = Math.sin(th);
	} else if (kx <= 1.05) {
		tmp = ky * (Math.sin(th) / kx);
	} else {
		tmp = ky * (th / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.26e-138:
		tmp = math.sin(th)
	elif kx <= 1.05:
		tmp = ky * (math.sin(th) / kx)
	else:
		tmp = ky * (th / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.26e-138)
		tmp = sin(th);
	elseif (kx <= 1.05)
		tmp = Float64(ky * Float64(sin(th) / kx));
	else
		tmp = Float64(ky * Float64(th / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.26e-138)
		tmp = sin(th);
	elseif (kx <= 1.05)
		tmp = ky * (sin(th) / kx);
	else
		tmp = ky * (th / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.26e-138], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 1.05], N[(ky * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 1.26e-138

   1. Initial program 89.1%

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

      1. unpow2 89.1%

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

      2. sqr-neg 89.1%

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

      3. sin-neg 89.1%

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

      4. sin-neg 89.1%

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

      5. unpow2 89.1%

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

      6. associate-*l/ 88.2%

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

      7. associate-/l* 89.0%

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

      8. unpow2 89.0%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 25.6%

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

   if 1.26e-138 < kx < 1.05000000000000004

   1. Initial program 99.9%

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

      1. unpow2 99.9%

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

      2. sqr-neg 99.9%

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

      3. sin-neg 99.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 99.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 99.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 98.3%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 99.6%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 99.6%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 50.8%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 52.3%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   7. Simplified 52.3%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   8. Taylor expanded in kx around 0 48.5%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]
   9. Step-by-step derivation

      1. associate-/l* 50.0%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{kx}} \]

   10. Simplified 50.0%

       \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{kx}} \]

   if 1.05000000000000004 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 99.4%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 99.4%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 99.4%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 99.4%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 99.4%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 99.3%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 99.4%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 99.4%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 33.3%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 33.3%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   7. Simplified 33.3%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   8. Taylor expanded in th around 0 15.7%

      \[\leadsto \color{blue}{\frac{ky \cdot th}{\sin kx}} \]
   9. Step-by-step derivation

      1. associate-/l* 15.7%

         \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   10. Simplified 15.7%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.26 \cdot 10^{-138}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 1.05:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 26.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 7.8e-53) (sin th) (* ky (/ th (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 7.8e-53) {
		tmp = sin(th);
	} else {
		tmp = ky * (th / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 7.8d-53) then
        tmp = sin(th)
    else
        tmp = ky * (th / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 7.8e-53) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (th / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 7.8e-53:
		tmp = math.sin(th)
	else:
		tmp = ky * (th / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 7.8e-53)
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(th / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 7.8e-53)
		tmp = sin(th);
	else
		tmp = ky * (th / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 7.8e-53], N[Sin[th], $MachinePrecision], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 7.8000000000000004e-53

   1. Initial program 90.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 90.1%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 90.1%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 90.1%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 90.1%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 90.1%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 89.0%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 90.0%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 90.0%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 25.6%

      \[\leadsto \color{blue}{\sin th} \]

   if 7.8000000000000004e-53 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 99.4%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 99.4%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ky around 0 39.3%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 39.3%

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   7. Simplified 39.3%

      \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]

   8. Taylor expanded in th around 0 18.2%

      \[\leadsto \color{blue}{\frac{ky \cdot th}{\sin kx}} \]
   9. Step-by-step derivation

      1. associate-/l* 18.2%

         \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]

   10. Simplified 18.2%

       \[\leadsto \color{blue}{ky \cdot \frac{th}{\sin kx}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 23.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \sin th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (sin th))

C

double code(double kx, double ky, double th) {
	return sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th);
}

Python

def code(kx, ky, th):
	return math.sin(th)

Julia

function code(kx, ky, th)
	return sin(th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[Sin[th], $MachinePrecision]

Derivation

1. Initial program 93.1%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. unpow2 93.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

   2. sqr-neg 93.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   3. sin-neg 93.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. sin-neg 93.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. unpow2 93.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

   6. associate-*l/ 92.3%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

   7. associate-/l* 93.0%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

   8. unpow2 93.0%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
4. Add Preprocessing

5. Taylor expanded in kx around 0 20.8%

   \[\leadsto \color{blue}{\sin th} \]

6. Final simplification 20.8%

   \[\leadsto \sin th \]
7. Add Preprocessing

Alternative 18: 13.3% accurate, 709.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return th;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.1%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. unpow2 93.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

   2. sqr-neg 93.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   3. sin-neg 93.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. sin-neg 93.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. unpow2 93.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

   6. associate-*l/ 92.3%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

   7. associate-/l* 93.0%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

   8. unpow2 93.0%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
4. Add Preprocessing

5. Taylor expanded in kx around 0 20.8%

   \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]

6. Taylor expanded in th around 0 10.3%

   \[\leadsto \color{blue}{th} \]

7. Final simplification 10.3%

   \[\leadsto th \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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)))