Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.3% → 99.7%

Time: 18.9s

Alternatives: 16

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% 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 92.8%

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

   1. remove-double-neg 92.8%

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

   2. sin-neg 92.8%

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

   3. neg-mul-1 92.8%

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

   4. *-commutative 92.8%

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

   5. associate-*l* 92.8%

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

   6. associate-*l/ 91.9%

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

   7. associate-/r/ 91.9%

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

   8. associate-*l/ 92.8%

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

   9. associate-/r/ 92.7%

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

   10. sin-neg 92.7%

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

   11. neg-mul-1 92.7%

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

   12. associate-/r* 92.7%

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

   13. associate-/r/ 92.8%

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

3. Simplified 99.7%

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

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: 43.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.05:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 10^{-44}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.05)
   (* (sin th) (fabs (/ ky (sin kx))))
   (if (<= (sin kx) 4e-142)
     (sin th)
     (if (<= (sin kx) 2e-66)
       (/ (sin ky) (/ kx (sin th)))
       (if (<= (sin kx) 1e-44) (sin th) (* (sin th) (/ (sin ky) (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.05) {
		tmp = sin(th) * fabs((ky / sin(kx)));
	} else if (sin(kx) <= 4e-142) {
		tmp = sin(th);
	} else if (sin(kx) <= 2e-66) {
		tmp = sin(ky) / (kx / sin(th));
	} else if (sin(kx) <= 1e-44) {
		tmp = sin(th);
	} else {
		tmp = 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(kx) <= (-0.05d0)) then
        tmp = sin(th) * abs((ky / sin(kx)))
    else if (sin(kx) <= 4d-142) then
        tmp = sin(th)
    else if (sin(kx) <= 2d-66) then
        tmp = sin(ky) / (kx / sin(th))
    else if (sin(kx) <= 1d-44) then
        tmp = sin(th)
    else
        tmp = 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(kx) <= -0.05) {
		tmp = Math.sin(th) * Math.abs((ky / Math.sin(kx)));
	} else if (Math.sin(kx) <= 4e-142) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 2e-66) {
		tmp = Math.sin(ky) / (kx / Math.sin(th));
	} else if (Math.sin(kx) <= 1e-44) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.05:
		tmp = math.sin(th) * math.fabs((ky / math.sin(kx)))
	elif math.sin(kx) <= 4e-142:
		tmp = math.sin(th)
	elif math.sin(kx) <= 2e-66:
		tmp = math.sin(ky) / (kx / math.sin(th))
	elif math.sin(kx) <= 1e-44:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.05)
		tmp = Float64(sin(th) * abs(Float64(ky / sin(kx))));
	elseif (sin(kx) <= 4e-142)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-66)
		tmp = Float64(sin(ky) / Float64(kx / sin(th)));
	elseif (sin(kx) <= 1e-44)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.05)
		tmp = sin(th) * abs((ky / sin(kx)));
	elseif (sin(kx) <= 4e-142)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-66)
		tmp = sin(ky) / (kx / sin(th));
	elseif (sin(kx) <= 1e-44)
		tmp = sin(th);
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.05], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-142], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-66], N[(N[Sin[ky], $MachinePrecision] / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-44], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.050000000000000003

   1. Initial program 99.3%

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

   3. Taylor expanded in ky around 0 10.5%

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

      1. associate-/l* 10.5%

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

      2. associate-/r/ 10.5%

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

   5. Simplified 10.5%

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

      1. add-sqr-sqrt 7.6%

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

      2. sqrt-unprod 14.8%

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

      3. pow2 14.8%

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

   7. Applied egg-rr 14.8%

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

      1. unpow2 14.8%

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

      2. rem-sqrt-square 18.0%

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

   9. Simplified 18.0%

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

   if -0.050000000000000003 < (sin.f64 kx) < 4.0000000000000002e-142 or 2e-66 < (sin.f64 kx) < 9.99999999999999953e-45

   1. Initial program 83.5%

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

   3. Taylor expanded in kx around 0 33.7%

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

   if 4.0000000000000002e-142 < (sin.f64 kx) < 2e-66

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0 48.0%

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

   4. Taylor expanded in kx around 0 48.0%

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

      1. associate-/l* 48.1%

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

   6. Simplified 48.1%

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

   if 9.99999999999999953e-45 < (sin.f64 kx)

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0 61.6%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.05:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 10^{-44}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.05:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 10^{-44}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.05)
   (* (sin th) (fabs (/ ky (sin kx))))
   (if (<= (sin kx) 4e-142)
     (sin th)
     (if (<= (sin kx) 2e-66)
       (/ (sin ky) (/ kx (sin th)))
       (if (<= (sin kx) 1e-44) (sin th) (* (sin ky) (/ (sin th) (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.05) {
		tmp = sin(th) * fabs((ky / sin(kx)));
	} else if (sin(kx) <= 4e-142) {
		tmp = sin(th);
	} else if (sin(kx) <= 2e-66) {
		tmp = sin(ky) / (kx / sin(th));
	} else if (sin(kx) <= 1e-44) {
		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) <= (-0.05d0)) then
        tmp = sin(th) * abs((ky / sin(kx)))
    else if (sin(kx) <= 4d-142) then
        tmp = sin(th)
    else if (sin(kx) <= 2d-66) then
        tmp = sin(ky) / (kx / sin(th))
    else if (sin(kx) <= 1d-44) 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) <= -0.05) {
		tmp = Math.sin(th) * Math.abs((ky / Math.sin(kx)));
	} else if (Math.sin(kx) <= 4e-142) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 2e-66) {
		tmp = Math.sin(ky) / (kx / Math.sin(th));
	} else if (Math.sin(kx) <= 1e-44) {
		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) <= -0.05:
		tmp = math.sin(th) * math.fabs((ky / math.sin(kx)))
	elif math.sin(kx) <= 4e-142:
		tmp = math.sin(th)
	elif math.sin(kx) <= 2e-66:
		tmp = math.sin(ky) / (kx / math.sin(th))
	elif math.sin(kx) <= 1e-44:
		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) <= -0.05)
		tmp = Float64(sin(th) * abs(Float64(ky / sin(kx))));
	elseif (sin(kx) <= 4e-142)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-66)
		tmp = Float64(sin(ky) / Float64(kx / sin(th)));
	elseif (sin(kx) <= 1e-44)
		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) <= -0.05)
		tmp = sin(th) * abs((ky / sin(kx)));
	elseif (sin(kx) <= 4e-142)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-66)
		tmp = sin(ky) / (kx / sin(th));
	elseif (sin(kx) <= 1e-44)
		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], -0.05], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-142], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-66], N[(N[Sin[ky], $MachinePrecision] / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-44], 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 4 regimes

2. if (sin.f64 kx) < -0.050000000000000003

   1. Initial program 99.3%

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

   3. Taylor expanded in ky around 0 10.5%

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

      1. associate-/l* 10.5%

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

      2. associate-/r/ 10.5%

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

   5. Simplified 10.5%

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

      1. add-sqr-sqrt 7.6%

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

      2. sqrt-unprod 14.8%

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

      3. pow2 14.8%

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

   7. Applied egg-rr 14.8%

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

      1. unpow2 14.8%

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

      2. rem-sqrt-square 18.0%

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

   9. Simplified 18.0%

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

   if -0.050000000000000003 < (sin.f64 kx) < 4.0000000000000002e-142 or 2e-66 < (sin.f64 kx) < 9.99999999999999953e-45

   1. Initial program 83.5%

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

   3. Taylor expanded in kx around 0 33.7%

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

   if 4.0000000000000002e-142 < (sin.f64 kx) < 2e-66

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0 48.0%

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

   4. Taylor expanded in kx around 0 48.0%

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

      1. associate-/l* 48.1%

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

   6. Simplified 48.1%

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

   if 9.99999999999999953e-45 < (sin.f64 kx)

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0 61.6%

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

   4. Taylor expanded in ky around inf 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*l/ 61.6%

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

   6. Simplified 61.6%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.05:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 10^{-44}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.05:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 10^{-44}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.05)
   (* (sin th) (fabs (/ ky (sin kx))))
   (if (<= (sin kx) 4e-142)
     (sin th)
     (if (<= (sin kx) 2e-66)
       (/ (sin ky) (/ kx (sin th)))
       (if (<= (sin kx) 1e-44)
         (/ (* (sin ky) (sin th)) (sin ky))
         (* (sin ky) (/ (sin th) (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.05) {
		tmp = sin(th) * fabs((ky / sin(kx)));
	} else if (sin(kx) <= 4e-142) {
		tmp = sin(th);
	} else if (sin(kx) <= 2e-66) {
		tmp = sin(ky) / (kx / sin(th));
	} else if (sin(kx) <= 1e-44) {
		tmp = (sin(ky) * sin(th)) / sin(ky);
	} 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) <= (-0.05d0)) then
        tmp = sin(th) * abs((ky / sin(kx)))
    else if (sin(kx) <= 4d-142) then
        tmp = sin(th)
    else if (sin(kx) <= 2d-66) then
        tmp = sin(ky) / (kx / sin(th))
    else if (sin(kx) <= 1d-44) then
        tmp = (sin(ky) * sin(th)) / sin(ky)
    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) <= -0.05) {
		tmp = Math.sin(th) * Math.abs((ky / Math.sin(kx)));
	} else if (Math.sin(kx) <= 4e-142) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 2e-66) {
		tmp = Math.sin(ky) / (kx / Math.sin(th));
	} else if (Math.sin(kx) <= 1e-44) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sin(ky);
	} 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) <= -0.05:
		tmp = math.sin(th) * math.fabs((ky / math.sin(kx)))
	elif math.sin(kx) <= 4e-142:
		tmp = math.sin(th)
	elif math.sin(kx) <= 2e-66:
		tmp = math.sin(ky) / (kx / math.sin(th))
	elif math.sin(kx) <= 1e-44:
		tmp = (math.sin(ky) * math.sin(th)) / math.sin(ky)
	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) <= -0.05)
		tmp = Float64(sin(th) * abs(Float64(ky / sin(kx))));
	elseif (sin(kx) <= 4e-142)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-66)
		tmp = Float64(sin(ky) / Float64(kx / sin(th)));
	elseif (sin(kx) <= 1e-44)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sin(ky));
	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) <= -0.05)
		tmp = sin(th) * abs((ky / sin(kx)));
	elseif (sin(kx) <= 4e-142)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-66)
		tmp = sin(ky) / (kx / sin(th));
	elseif (sin(kx) <= 1e-44)
		tmp = (sin(ky) * sin(th)) / sin(ky);
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.05], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-142], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-66], N[(N[Sin[ky], $MachinePrecision] / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-44], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 kx) < -0.050000000000000003

   1. Initial program 99.3%

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

   3. Taylor expanded in ky around 0 10.5%

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

      1. associate-/l* 10.5%

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

      2. associate-/r/ 10.5%

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

   5. Simplified 10.5%

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

      1. add-sqr-sqrt 7.6%

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

      2. sqrt-unprod 14.8%

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

      3. pow2 14.8%

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

   7. Applied egg-rr 14.8%

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

      1. unpow2 14.8%

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

      2. rem-sqrt-square 18.0%

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

   9. Simplified 18.0%

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

   if -0.050000000000000003 < (sin.f64 kx) < 4.0000000000000002e-142

   1. Initial program 83.0%

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

   3. Taylor expanded in kx around 0 33.7%

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

   if 4.0000000000000002e-142 < (sin.f64 kx) < 2e-66

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0 48.0%

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

   4. Taylor expanded in kx around 0 48.0%

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

      1. associate-/l* 48.1%

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

   6. Simplified 48.1%

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

   if 2e-66 < (sin.f64 kx) < 9.99999999999999953e-45

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)\right)} \]

      2. expm1-udef 68.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)} - 1} \]

   4. Applied egg-rr 68.4%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*l/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in kx around 0 35.1%

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

   if 9.99999999999999953e-45 < (sin.f64 kx)

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0 61.6%

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

   4. Taylor expanded in ky around inf 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*l/ 61.6%

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

   6. Simplified 61.6%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.05:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 10^{-44}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 39.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.05:\\ \;\;\;\;\left|\sin ky \cdot \frac{th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 10^{-39}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.05)
   (fabs (* (sin ky) (/ th (sin kx))))
   (if (<= (sin kx) 4e-142)
     (sin th)
     (if (<= (sin kx) 2e-66)
       (/ (sin ky) (/ kx (sin th)))
       (if (<= (sin kx) 1e-39) (sin th) (* (sin th) (/ ky (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.05) {
		tmp = fabs((sin(ky) * (th / sin(kx))));
	} else if (sin(kx) <= 4e-142) {
		tmp = sin(th);
	} else if (sin(kx) <= 2e-66) {
		tmp = sin(ky) / (kx / sin(th));
	} else if (sin(kx) <= 1e-39) {
		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) <= (-0.05d0)) then
        tmp = abs((sin(ky) * (th / sin(kx))))
    else if (sin(kx) <= 4d-142) then
        tmp = sin(th)
    else if (sin(kx) <= 2d-66) then
        tmp = sin(ky) / (kx / sin(th))
    else if (sin(kx) <= 1d-39) 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) <= -0.05) {
		tmp = Math.abs((Math.sin(ky) * (th / Math.sin(kx))));
	} else if (Math.sin(kx) <= 4e-142) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 2e-66) {
		tmp = Math.sin(ky) / (kx / Math.sin(th));
	} else if (Math.sin(kx) <= 1e-39) {
		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) <= -0.05:
		tmp = math.fabs((math.sin(ky) * (th / math.sin(kx))))
	elif math.sin(kx) <= 4e-142:
		tmp = math.sin(th)
	elif math.sin(kx) <= 2e-66:
		tmp = math.sin(ky) / (kx / math.sin(th))
	elif math.sin(kx) <= 1e-39:
		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) <= -0.05)
		tmp = abs(Float64(sin(ky) * Float64(th / sin(kx))));
	elseif (sin(kx) <= 4e-142)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-66)
		tmp = Float64(sin(ky) / Float64(kx / sin(th)));
	elseif (sin(kx) <= 1e-39)
		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) <= -0.05)
		tmp = abs((sin(ky) * (th / sin(kx))));
	elseif (sin(kx) <= 4e-142)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-66)
		tmp = sin(ky) / (kx / sin(th));
	elseif (sin(kx) <= 1e-39)
		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], -0.05], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-142], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-66], N[(N[Sin[ky], $MachinePrecision] / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-39], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.050000000000000003

   1. Initial program 99.3%

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

   3. Taylor expanded in ky around 0 10.1%

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

   4. Taylor expanded in th around 0 11.0%

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

      1. add-sqr-sqrt 8.4%

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

      2. sqrt-unprod 15.1%

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

      3. pow2 15.1%

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

      4. associate-/l* 15.1%

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

      5. div-inv 15.1%

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

      6. clear-num 15.1%

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

   6. Applied egg-rr 15.1%

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

      1. unpow2 15.1%

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

      2. rem-sqrt-square 20.3%

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

      3. associate-*r/ 20.3%

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

      4. associate-*l/ 20.3%

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

      5. *-commutative 20.3%

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

   8. Simplified 20.3%

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

   if -0.050000000000000003 < (sin.f64 kx) < 4.0000000000000002e-142 or 2e-66 < (sin.f64 kx) < 9.99999999999999929e-40

   1. Initial program 83.7%

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

   3. Taylor expanded in kx around 0 33.4%

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

   if 4.0000000000000002e-142 < (sin.f64 kx) < 2e-66

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0 48.0%

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

   4. Taylor expanded in kx around 0 48.0%

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

      1. associate-/l* 48.1%

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

   6. Simplified 48.1%

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

   if 9.99999999999999929e-40 < (sin.f64 kx)

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0 54.4%

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

      1. associate-/l* 55.7%

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

      2. associate-/r/ 55.7%

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

   5. Simplified 55.7%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.05:\\ \;\;\;\;\left|\sin ky \cdot \frac{th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 10^{-39}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.05:\\ \;\;\;\;th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 10^{-39}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.05)
   (* th (/ (sin ky) (fabs (sin kx))))
   (if (<= (sin kx) 4e-142)
     (sin th)
     (if (<= (sin kx) 2e-66)
       (/ (sin ky) (/ kx (sin th)))
       (if (<= (sin kx) 1e-39) (sin th) (* (sin th) (/ ky (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.05) {
		tmp = th * (sin(ky) / fabs(sin(kx)));
	} else if (sin(kx) <= 4e-142) {
		tmp = sin(th);
	} else if (sin(kx) <= 2e-66) {
		tmp = sin(ky) / (kx / sin(th));
	} else if (sin(kx) <= 1e-39) {
		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) <= (-0.05d0)) then
        tmp = th * (sin(ky) / abs(sin(kx)))
    else if (sin(kx) <= 4d-142) then
        tmp = sin(th)
    else if (sin(kx) <= 2d-66) then
        tmp = sin(ky) / (kx / sin(th))
    else if (sin(kx) <= 1d-39) 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) <= -0.05) {
		tmp = th * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	} else if (Math.sin(kx) <= 4e-142) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 2e-66) {
		tmp = Math.sin(ky) / (kx / Math.sin(th));
	} else if (Math.sin(kx) <= 1e-39) {
		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) <= -0.05:
		tmp = th * (math.sin(ky) / math.fabs(math.sin(kx)))
	elif math.sin(kx) <= 4e-142:
		tmp = math.sin(th)
	elif math.sin(kx) <= 2e-66:
		tmp = math.sin(ky) / (kx / math.sin(th))
	elif math.sin(kx) <= 1e-39:
		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) <= -0.05)
		tmp = Float64(th * Float64(sin(ky) / abs(sin(kx))));
	elseif (sin(kx) <= 4e-142)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-66)
		tmp = Float64(sin(ky) / Float64(kx / sin(th)));
	elseif (sin(kx) <= 1e-39)
		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) <= -0.05)
		tmp = th * (sin(ky) / abs(sin(kx)));
	elseif (sin(kx) <= 4e-142)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-66)
		tmp = sin(ky) / (kx / sin(th));
	elseif (sin(kx) <= 1e-39)
		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], -0.05], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-142], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-66], N[(N[Sin[ky], $MachinePrecision] / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-39], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.050000000000000003

   1. Initial program 99.3%

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

   3. Taylor expanded in ky around 0 10.1%

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

      1. add-sqr-sqrt 7.5%

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

      2. sqrt-unprod 20.0%

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

      3. pow2 20.0%

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

   5. Applied egg-rr 20.0%

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

      1. unpow2 20.0%

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

      2. rem-sqrt-square 22.9%

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

   7. Simplified 22.9%

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

   8. Taylor expanded in th around 0 18.7%

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

      1. fabs-div 18.7%

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

      2. rem-square-sqrt 10.6%

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

      3. fabs-sqr 10.6%

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

      4. rem-square-sqrt 30.1%

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

   10. Simplified 30.1%

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

   if -0.050000000000000003 < (sin.f64 kx) < 4.0000000000000002e-142 or 2e-66 < (sin.f64 kx) < 9.99999999999999929e-40

   1. Initial program 83.7%

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

   3. Taylor expanded in kx around 0 33.4%

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

   if 4.0000000000000002e-142 < (sin.f64 kx) < 2e-66

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0 48.0%

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

   4. Taylor expanded in kx around 0 48.0%

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

      1. associate-/l* 48.1%

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

   6. Simplified 48.1%

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

   if 9.99999999999999929e-40 < (sin.f64 kx)

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0 54.4%

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

      1. associate-/l* 55.7%

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

      2. associate-/r/ 55.7%

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

   5. Simplified 55.7%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.05:\\ \;\;\;\;th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 10^{-39}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

   1. expm1-log1p-u 92.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)\right)} \]

   2. expm1-udef 43.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)} - 1} \]

4. Applied egg-rr 46.0%

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

   1. expm1-def 99.6%

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

   2. expm1-log1p 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*r/ 95.2%

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

   5. associate-*l/ 99.6%

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

   6. *-commutative 99.6%

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

   7. hypot-def 92.7%

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

   8. unpow2 92.7%

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

   9. unpow2 92.7%

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

   10. +-commutative 92.7%

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

   11. unpow2 92.7%

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

   12. unpow2 92.7%

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

   13. hypot-def 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 8: 43.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ \mathbf{if}\;ky \leq 1.05 \cdot 10^{-221}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;ky \leq 0.0038:\\ \;\;\;\;\frac{ky \cdot \sin th}{t\_1}\\ \mathbf{elif}\;ky \leq 5.35 \cdot 10^{+179} \lor \neg \left(ky \leq 1.3 \cdot 10^{+220}\right):\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin ky\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot th}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky))))
   (if (<= ky 1.05e-221)
     (* (sin th) (fabs (/ ky (sin kx))))
     (if (<= ky 0.0038)
       (/ (* ky (sin th)) t_1)
       (if (or (<= ky 5.35e+179) (not (<= ky 1.3e+220)))
         (* (sin th) (/ (sin ky) (fabs (sin ky))))
         (/ (* (sin ky) th) t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double tmp;
	if (ky <= 1.05e-221) {
		tmp = sin(th) * fabs((ky / sin(kx)));
	} else if (ky <= 0.0038) {
		tmp = (ky * sin(th)) / t_1;
	} else if ((ky <= 5.35e+179) || !(ky <= 1.3e+220)) {
		tmp = sin(th) * (sin(ky) / fabs(sin(ky)));
	} else {
		tmp = (sin(ky) * th) / t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(kx), Math.sin(ky));
	double tmp;
	if (ky <= 1.05e-221) {
		tmp = Math.sin(th) * Math.abs((ky / Math.sin(kx)));
	} else if (ky <= 0.0038) {
		tmp = (ky * Math.sin(th)) / t_1;
	} else if ((ky <= 5.35e+179) || !(ky <= 1.3e+220)) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(ky)));
	} else {
		tmp = (Math.sin(ky) * th) / t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if ky <= 1.05e-221:
		tmp = math.sin(th) * math.fabs((ky / math.sin(kx)))
	elif ky <= 0.0038:
		tmp = (ky * math.sin(th)) / t_1
	elif (ky <= 5.35e+179) or not (ky <= 1.3e+220):
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(ky)))
	else:
		tmp = (math.sin(ky) * th) / t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	tmp = 0.0
	if (ky <= 1.05e-221)
		tmp = Float64(sin(th) * abs(Float64(ky / sin(kx))));
	elseif (ky <= 0.0038)
		tmp = Float64(Float64(ky * sin(th)) / t_1);
	elseif ((ky <= 5.35e+179) || !(ky <= 1.3e+220))
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(ky))));
	else
		tmp = Float64(Float64(sin(ky) * th) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (ky <= 1.05e-221)
		tmp = sin(th) * abs((ky / sin(kx)));
	elseif (ky <= 0.0038)
		tmp = (ky * sin(th)) / t_1;
	elseif ((ky <= 5.35e+179) || ~((ky <= 1.3e+220)))
		tmp = sin(th) * (sin(ky) / abs(sin(ky)));
	else
		tmp = (sin(ky) * th) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[ky, 1.05e-221], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 0.0038], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[Or[LessEqual[ky, 5.35e+179], N[Not[LessEqual[ky, 1.3e+220]], $MachinePrecision]], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if ky < 1.05e-221

   1. Initial program 90.5%

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

   3. Taylor expanded in ky around 0 26.5%

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

      1. associate-/l* 29.1%

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

      2. associate-/r/ 29.1%

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

   5. Simplified 29.1%

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

      1. add-sqr-sqrt 10.5%

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

      2. sqrt-unprod 15.7%

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

      3. pow2 15.7%

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

   7. Applied egg-rr 15.7%

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

      1. unpow2 15.7%

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

      2. rem-sqrt-square 18.7%

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

   9. Simplified 18.7%

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

   if 1.05e-221 < ky < 0.00379999999999999999

   1. Initial program 91.5%

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

      1. expm1-log1p-u 91.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)\right)} \]

      2. expm1-udef 24.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)} - 1} \]

   4. Applied egg-rr 27.9%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

      3. associate-*l/ 92.0%

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

   6. Simplified 92.0%

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

   7. Taylor expanded in ky around 0 90.6%

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

   if 0.00379999999999999999 < ky < 5.35000000000000012e179 or 1.29999999999999997e220 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. unpow2 99.7%

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

      3. hypot-def 99.8%

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

      4. expm1-log1p-u 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in kx around 0 26.2%

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

      1. expm1-log1p-u 26.3%

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

      2. add-sqr-sqrt 24.7%

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

      3. fabs-sqr 24.7%

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

      4. add-sqr-sqrt 63.1%

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

   7. Applied egg-rr 63.1%

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

   if 5.35000000000000012e179 < ky < 1.29999999999999997e220

   1. Initial program 99.2%

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

      1. expm1-log1p-u 99.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)\right)} \]

      2. expm1-udef 39.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)} - 1} \]

   4. Applied egg-rr 39.7%

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

      1. expm1-def 99.2%

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

      2. expm1-log1p 99.2%

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

      3. associate-*l/ 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in th around 0 74.5%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 1.05 \cdot 10^{-221}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;ky \leq 0.0038:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{elif}\;ky \leq 5.35 \cdot 10^{+179} \lor \neg \left(ky \leq 1.3 \cdot 10^{+220}\right):\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin ky\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 4.2 \cdot 10^{-142}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 8 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;kx \leq 5.9 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 4.2e-142)
   (sin th)
   (if (<= kx 8e-66)
     (/ (sin ky) (/ kx (sin th)))
     (if (<= kx 5.9e-39)
       (/ (* (sin ky) (sin th)) (sin ky))
       (* (sin th) (fabs (/ (sin ky) (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.2e-142) {
		tmp = sin(th);
	} else if (kx <= 8e-66) {
		tmp = sin(ky) / (kx / sin(th));
	} else if (kx <= 5.9e-39) {
		tmp = (sin(ky) * sin(th)) / sin(ky);
	} else {
		tmp = sin(th) * fabs((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 (kx <= 4.2d-142) then
        tmp = sin(th)
    else if (kx <= 8d-66) then
        tmp = sin(ky) / (kx / sin(th))
    else if (kx <= 5.9d-39) then
        tmp = (sin(ky) * sin(th)) / sin(ky)
    else
        tmp = sin(th) * abs((sin(ky) / sin(kx)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.2e-142) {
		tmp = Math.sin(th);
	} else if (kx <= 8e-66) {
		tmp = Math.sin(ky) / (kx / Math.sin(th));
	} else if (kx <= 5.9e-39) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sin(ky);
	} else {
		tmp = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 4.2e-142:
		tmp = math.sin(th)
	elif kx <= 8e-66:
		tmp = math.sin(ky) / (kx / math.sin(th))
	elif kx <= 5.9e-39:
		tmp = (math.sin(ky) * math.sin(th)) / math.sin(ky)
	else:
		tmp = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 4.2e-142)
		tmp = sin(th);
	elseif (kx <= 8e-66)
		tmp = Float64(sin(ky) / Float64(kx / sin(th)));
	elseif (kx <= 5.9e-39)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sin(ky));
	else
		tmp = Float64(sin(th) * abs(Float64(sin(ky) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 4.2e-142)
		tmp = sin(th);
	elseif (kx <= 8e-66)
		tmp = sin(ky) / (kx / sin(th));
	elseif (kx <= 5.9e-39)
		tmp = (sin(ky) * sin(th)) / sin(ky);
	else
		tmp = sin(th) * abs((sin(ky) / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 4.2e-142], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 8e-66], N[(N[Sin[ky], $MachinePrecision] / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 5.9e-39], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if kx < 4.1999999999999999e-142

   1. Initial program 89.4%

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

   3. Taylor expanded in kx around 0 23.5%

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

   if 4.1999999999999999e-142 < kx < 7.9999999999999998e-66

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0 48.0%

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

   4. Taylor expanded in kx around 0 48.0%

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

      1. associate-/l* 48.1%

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

   6. Simplified 48.1%

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

   if 7.9999999999999998e-66 < kx < 5.8999999999999998e-39

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)\right)} \]

      2. expm1-udef 52.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)} - 1} \]

   4. Applied egg-rr 52.6%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*l/ 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in kx around 0 27.2%

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

   if 5.8999999999999998e-39 < kx

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0 41.6%

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

      1. add-sqr-sqrt 18.2%

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

      2. sqrt-unprod 29.3%

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

      3. pow2 29.3%

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

   5. Applied egg-rr 29.3%

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

      1. unpow2 29.3%

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

      2. rem-sqrt-square 33.2%

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

   7. Simplified 33.2%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 4.2 \cdot 10^{-142}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 8 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;kx \leq 5.9 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 9.2 \cdot 10^{-222}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;ky \leq 0.0025:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin ky\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 9.2e-222)
   (* (sin th) (fabs (/ ky (sin kx))))
   (if (<= ky 0.0025)
     (/ (* ky (sin th)) (hypot (sin kx) (sin ky)))
     (* (sin th) (/ (sin ky) (fabs (sin ky)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 9.2e-222) {
		tmp = sin(th) * fabs((ky / sin(kx)));
	} else if (ky <= 0.0025) {
		tmp = (ky * sin(th)) / hypot(sin(kx), sin(ky));
	} else {
		tmp = sin(th) * (sin(ky) / fabs(sin(ky)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 9.2e-222) {
		tmp = Math.sin(th) * Math.abs((ky / Math.sin(kx)));
	} else if (ky <= 0.0025) {
		tmp = (ky * Math.sin(th)) / Math.hypot(Math.sin(kx), Math.sin(ky));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 9.2e-222:
		tmp = math.sin(th) * math.fabs((ky / math.sin(kx)))
	elif ky <= 0.0025:
		tmp = (ky * math.sin(th)) / math.hypot(math.sin(kx), math.sin(ky))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 9.2e-222)
		tmp = Float64(sin(th) * abs(Float64(ky / sin(kx))));
	elseif (ky <= 0.0025)
		tmp = Float64(Float64(ky * sin(th)) / hypot(sin(kx), sin(ky)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 9.2e-222)
		tmp = sin(th) * abs((ky / sin(kx)));
	elseif (ky <= 0.0025)
		tmp = (ky * sin(th)) / hypot(sin(kx), sin(ky));
	else
		tmp = sin(th) * (sin(ky) / abs(sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 9.2e-222], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 0.0025], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 9.2000000000000005e-222

   1. Initial program 90.5%

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

   3. Taylor expanded in ky around 0 26.5%

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

      1. associate-/l* 29.1%

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

      2. associate-/r/ 29.1%

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

   5. Simplified 29.1%

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

      1. add-sqr-sqrt 10.5%

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

      2. sqrt-unprod 15.7%

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

      3. pow2 15.7%

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

   7. Applied egg-rr 15.7%

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

      1. unpow2 15.7%

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

      2. rem-sqrt-square 18.7%

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

   9. Simplified 18.7%

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

   if 9.2000000000000005e-222 < ky < 0.00250000000000000005

   1. Initial program 91.5%

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

      1. expm1-log1p-u 91.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)\right)} \]

      2. expm1-udef 24.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)} - 1} \]

   4. Applied egg-rr 27.9%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

      3. associate-*l/ 92.0%

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

   6. Simplified 92.0%

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

   7. Taylor expanded in ky around 0 90.6%

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

   if 0.00250000000000000005 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. unpow2 99.6%

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

      3. hypot-def 99.7%

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

      4. expm1-log1p-u 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in kx around 0 24.1%

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

      1. expm1-log1p-u 24.2%

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

      2. add-sqr-sqrt 22.7%

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

      3. fabs-sqr 22.7%

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

      4. add-sqr-sqrt 58.1%

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

   7. Applied egg-rr 58.1%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 9.2 \cdot 10^{-222}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;ky \leq 0.0025:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin ky\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.00195:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin ky\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.00195)
   (* (sin th) (fabs (/ (sin ky) (sin kx))))
   (* (sin th) (/ (sin ky) (fabs (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.00195) {
		tmp = sin(th) * fabs((sin(ky) / sin(kx)));
	} else {
		tmp = sin(th) * (sin(ky) / fabs(sin(ky)));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 0.00195d0) then
        tmp = sin(th) * abs((sin(ky) / sin(kx)))
    else
        tmp = sin(th) * (sin(ky) / abs(sin(ky)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.00195) {
		tmp = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.00195:
		tmp = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.00195)
		tmp = Float64(sin(th) * abs(Float64(sin(ky) / sin(kx))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.00195)
		tmp = sin(th) * abs((sin(ky) / sin(kx)));
	else
		tmp = sin(th) * (sin(ky) / abs(sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.00195], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.0019499999999999999

   1. Initial program 90.7%

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

   3. Taylor expanded in ky around 0 33.2%

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

      1. add-sqr-sqrt 16.3%

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

      2. sqrt-unprod 25.2%

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

      3. pow2 25.2%

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

   5. Applied egg-rr 25.2%

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

      1. unpow2 25.2%

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

      2. rem-sqrt-square 28.5%

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

   7. Simplified 28.5%

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

   if 0.0019499999999999999 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. unpow2 99.6%

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

      3. hypot-def 99.7%

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

      4. expm1-log1p-u 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in kx around 0 24.1%

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

      1. expm1-log1p-u 24.2%

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

      2. add-sqr-sqrt 22.7%

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

      3. fabs-sqr 22.7%

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

      4. add-sqr-sqrt 58.1%

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

   7. Applied egg-rr 58.1%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.00195:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin ky\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 32.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if ky < 2e-3

   1. Initial program 90.7%

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

   3. Taylor expanded in ky around 0 28.9%

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

      1. associate-/l* 31.5%

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

      2. associate-/r/ 31.5%

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

   5. Simplified 31.5%

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

      1. add-sqr-sqrt 15.6%

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

      2. sqrt-unprod 23.5%

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

      3. pow2 23.5%

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

   7. Applied egg-rr 23.5%

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

      1. unpow2 23.5%

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

      2. rem-sqrt-square 26.9%

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

   9. Simplified 26.9%

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

   if 2e-3 < ky

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around 0 24.2%

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

4. Final simplification 26.2%

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

Alternative 13: 32.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 9.5 \cdot 10^{-46}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 9.5e-46) (* (sin th) (/ ky (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 9.5e-46) {
		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 (ky <= 9.5d-46) 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 (ky <= 9.5e-46) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 9.5e-46:
		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 (ky <= 9.5e-46)
		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 (ky <= 9.5e-46)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 9.5e-46], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 9.49999999999999993e-46

   1. Initial program 90.2%

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

   3. Taylor expanded in ky around 0 28.5%

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

      1. associate-/l* 31.2%

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

      2. associate-/r/ 31.2%

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

   5. Simplified 31.2%

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

   if 9.49999999999999993e-46 < ky

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around 0 28.6%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 9.5 \cdot 10^{-46}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 4.8 \cdot 10^{-109}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 4.8e-109) (* (sin th) (/ ky kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.8e-109) {
		tmp = sin(th) * (ky / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 4.8d-109) then
        tmp = sin(th) * (ky / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.8e-109) {
		tmp = Math.sin(th) * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 4.8e-109:
		tmp = math.sin(th) * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 4.8e-109)
		tmp = Float64(sin(th) * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 4.8e-109)
		tmp = sin(th) * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 4.8e-109], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 4.79999999999999977e-109

   1. Initial program 89.6%

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

   3. Taylor expanded in ky around 0 26.8%

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

      1. associate-/l* 29.7%

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

      2. associate-/r/ 29.7%

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

   5. Simplified 29.7%

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

   6. Taylor expanded in kx around 0 21.6%

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

   if 4.79999999999999977e-109 < ky

   1. Initial program 99.7%

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

   3. Taylor expanded in kx around 0 29.1%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 4.8 \cdot 10^{-109}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 23.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.25 \cdot 10^{-122}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.25e-122) (/ ky (/ kx th)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.25e-122) {
		tmp = ky / (kx / th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 1.25d-122) then
        tmp = ky / (kx / th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.25e-122) {
		tmp = ky / (kx / th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.25e-122:
		tmp = ky / (kx / th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.25e-122)
		tmp = Float64(ky / Float64(kx / th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.25e-122)
		tmp = ky / (kx / th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.25e-122], N[(ky / N[(kx / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.25e-122

   1. Initial program 89.3%

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

   3. Taylor expanded in ky around 0 26.8%

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

      1. associate-/l* 29.7%

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

      2. associate-/r/ 29.7%

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

   5. Simplified 29.7%

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

   6. Taylor expanded in kx around 0 21.4%

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

   7. Taylor expanded in th around 0 14.6%

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

      1. associate-/l* 17.6%

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

   9. Simplified 17.6%

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

   if 1.25e-122 < ky

   1. Initial program 99.7%

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

   3. Taylor expanded in kx around 0 29.3%

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

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 1.25 \cdot 10^{-122}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 13.3% accurate, 141.8× speedup

Math

\[\begin{array}{l} \\ \frac{ky}{\frac{kx}{th}} \end{array} \]

FPCore

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

C

double code(double kx, double ky, double th) {
	return ky / (kx / 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 = ky / (kx / th)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[(ky / N[(kx / th), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.8%

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

3. Taylor expanded in ky around 0 22.8%

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

   1. associate-/l* 24.8%

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

   2. associate-/r/ 24.8%

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

5. Simplified 24.8%

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

6. Taylor expanded in kx around 0 16.8%

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

7. Taylor expanded in th around 0 11.7%

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

   1. associate-/l* 13.8%

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

9. Simplified 13.8%

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

10. Final simplification 13.8%

    \[\leadsto \frac{ky}{\frac{kx}{th}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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)))