Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.3% → 99.7%

Time: 14.8s

Alternatives: 14

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

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1. Add Preprocessing

Alternative 2: 77.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\frac{\sin ky \cdot th}{t_1}\\ \mathbf{elif}\;\sin ky \leq 0.004:\\ \;\;\;\;\frac{\frac{\sin th}{t_1}}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky))))
   (if (<= (sin ky) -0.005)
     (/ (* (sin ky) th) t_1)
     (if (<= (sin ky) 0.004)
       (/ (/ (sin th) t_1) (+ (* ky 0.16666666666666666) (/ 1.0 ky)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double tmp;
	if (sin(ky) <= -0.005) {
		tmp = (sin(ky) * th) / t_1;
	} else if (sin(ky) <= 0.004) {
		tmp = (sin(th) / t_1) / ((ky * 0.16666666666666666) + (1.0 / ky));
	} else {
		tmp = sin(th);
	}
	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 (Math.sin(ky) <= -0.005) {
		tmp = (Math.sin(ky) * th) / t_1;
	} else if (Math.sin(ky) <= 0.004) {
		tmp = (Math.sin(th) / t_1) / ((ky * 0.16666666666666666) + (1.0 / ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if math.sin(ky) <= -0.005:
		tmp = (math.sin(ky) * th) / t_1
	elif math.sin(ky) <= 0.004:
		tmp = (math.sin(th) / t_1) / ((ky * 0.16666666666666666) + (1.0 / ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	tmp = 0.0
	if (sin(ky) <= -0.005)
		tmp = Float64(Float64(sin(ky) * th) / t_1);
	elseif (sin(ky) <= 0.004)
		tmp = Float64(Float64(sin(th) / t_1) / Float64(Float64(ky * 0.16666666666666666) + Float64(1.0 / ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (sin(ky) <= -0.005)
		tmp = (sin(ky) * th) / t_1;
	elseif (sin(ky) <= 0.004)
		tmp = (sin(th) / t_1) / ((ky * 0.16666666666666666) + (1.0 / ky));
	else
		tmp = sin(th);
	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[N[Sin[ky], $MachinePrecision], -0.005], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.004], N[(N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision] / N[(N[(ky * 0.16666666666666666), $MachinePrecision] + N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

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1. Add Preprocessing

Alternative 3: 77.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\frac{\sin ky \cdot th}{t_1}\\ \mathbf{elif}\;\sin ky \leq 10^{-7}:\\ \;\;\;\;\frac{\frac{\sin th}{t_1}}{\frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky))))
   (if (<= (sin ky) -0.005)
     (/ (* (sin ky) th) t_1)
     (if (<= (sin ky) 1e-7) (/ (/ (sin th) t_1) (/ 1.0 ky)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double tmp;
	if (sin(ky) <= -0.005) {
		tmp = (sin(ky) * th) / t_1;
	} else if (sin(ky) <= 1e-7) {
		tmp = (sin(th) / t_1) / (1.0 / ky);
	} else {
		tmp = sin(th);
	}
	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 (Math.sin(ky) <= -0.005) {
		tmp = (Math.sin(ky) * th) / t_1;
	} else if (Math.sin(ky) <= 1e-7) {
		tmp = (Math.sin(th) / t_1) / (1.0 / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if math.sin(ky) <= -0.005:
		tmp = (math.sin(ky) * th) / t_1
	elif math.sin(ky) <= 1e-7:
		tmp = (math.sin(th) / t_1) / (1.0 / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	tmp = 0.0
	if (sin(ky) <= -0.005)
		tmp = Float64(Float64(sin(ky) * th) / t_1);
	elseif (sin(ky) <= 1e-7)
		tmp = Float64(Float64(sin(th) / t_1) / Float64(1.0 / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (sin(ky) <= -0.005)
		tmp = (sin(ky) * th) / t_1;
	elseif (sin(ky) <= 1e-7)
		tmp = (sin(th) / t_1) / (1.0 / ky);
	else
		tmp = sin(th);
	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[N[Sin[ky], $MachinePrecision], -0.005], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-7], N[(N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision] / N[(1.0 / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

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1. Add Preprocessing

Alternative 4: 51.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-282}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-7}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-282)
   (* (sin ky) (/ (sin th) (sin kx)))
   (if (<= (sin ky) 1e-7)
     (/ (* ky (sin th)) (hypot (sin kx) (sin ky)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-282) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (sin(ky) <= 1e-7) {
		tmp = (ky * sin(th)) / hypot(sin(kx), sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 1e-282) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(ky) <= 1e-7) {
		tmp = (ky * Math.sin(th)) / Math.hypot(Math.sin(kx), Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-282:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif math.sin(ky) <= 1e-7:
		tmp = (ky * math.sin(th)) / math.hypot(math.sin(kx), math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-282)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (sin(ky) <= 1e-7)
		tmp = Float64(Float64(ky * sin(th)) / hypot(sin(kx), sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-282)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (sin(ky) <= 1e-7)
		tmp = (ky * sin(th)) / hypot(sin(kx), sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-282], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-7], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 5: 73.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\frac{\sin ky \cdot th}{t_1}\\ \mathbf{elif}\;\sin ky \leq 10^{-7}:\\ \;\;\;\;\frac{ky \cdot \sin th}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky))))
   (if (<= (sin ky) -0.005)
     (/ (* (sin ky) th) t_1)
     (if (<= (sin ky) 1e-7) (/ (* ky (sin th)) t_1) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double tmp;
	if (sin(ky) <= -0.005) {
		tmp = (sin(ky) * th) / t_1;
	} else if (sin(ky) <= 1e-7) {
		tmp = (ky * sin(th)) / t_1;
	} else {
		tmp = sin(th);
	}
	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 (Math.sin(ky) <= -0.005) {
		tmp = (Math.sin(ky) * th) / t_1;
	} else if (Math.sin(ky) <= 1e-7) {
		tmp = (ky * Math.sin(th)) / t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if math.sin(ky) <= -0.005:
		tmp = (math.sin(ky) * th) / t_1
	elif math.sin(ky) <= 1e-7:
		tmp = (ky * math.sin(th)) / t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	tmp = 0.0
	if (sin(ky) <= -0.005)
		tmp = Float64(Float64(sin(ky) * th) / t_1);
	elseif (sin(ky) <= 1e-7)
		tmp = Float64(Float64(ky * sin(th)) / t_1);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (sin(ky) <= -0.005)
		tmp = (sin(ky) * th) / t_1;
	elseif (sin(ky) <= 1e-7)
		tmp = (ky * sin(th)) / t_1;
	else
		tmp = sin(th);
	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[N[Sin[ky], $MachinePrecision], -0.005], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-7], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 42.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.07:\\ \;\;\;\;\sin th \cdot \sqrt{{\left(\frac{ky}{\sin kx}\right)}^{2}}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-68}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.07)
   (* (sin th) (sqrt (pow (/ ky (sin kx)) 2.0)))
   (if (<= (sin kx) 5e-68) (sin th) (/ (sin ky) (/ (sin kx) (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.07) {
		tmp = sin(th) * sqrt(pow((ky / sin(kx)), 2.0));
	} else if (sin(kx) <= 5e-68) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) / (sin(kx) / sin(th));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.07d0)) then
        tmp = sin(th) * sqrt(((ky / sin(kx)) ** 2.0d0))
    else if (sin(kx) <= 5d-68) then
        tmp = sin(th)
    else
        tmp = sin(ky) / (sin(kx) / sin(th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.07) {
		tmp = Math.sin(th) * Math.sqrt(Math.pow((ky / Math.sin(kx)), 2.0));
	} else if (Math.sin(kx) <= 5e-68) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.07:
		tmp = math.sin(th) * math.sqrt(math.pow((ky / math.sin(kx)), 2.0))
	elif math.sin(kx) <= 5e-68:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.07)
		tmp = Float64(sin(th) * sqrt((Float64(ky / sin(kx)) ^ 2.0)));
	elseif (sin(kx) <= 5e-68)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.07)
		tmp = sin(th) * sqrt(((ky / sin(kx)) ^ 2.0));
	elseif (sin(kx) <= 5e-68)
		tmp = sin(th);
	else
		tmp = sin(ky) / (sin(kx) / sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.07], N[(N[Sin[th], $MachinePrecision] * N[Sqrt[N[Power[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-68], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

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Alternative 7: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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Alternative 8: 41.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-53}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-53) (* (sin ky) (/ (sin th) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-53) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 1d-53) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 1e-53) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-53:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-53)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-53)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-53], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

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Alternative 9: 33.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 3.1 \cdot 10^{-51}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 3.1e-51) (* ky (/ (sin th) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3.1e-51) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 3.1d-51) then
        tmp = ky * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3.1e-51) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 3.1e-51:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 3.1e-51)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 3.1e-51)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 3.1e-51], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

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Alternative 10: 26.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.2 \cdot 10^{-150}:\\ \;\;\;\;ky \cdot \frac{-\sin th}{-kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.2e-150) (* ky (/ (- (sin th)) (- kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.2e-150) {
		tmp = ky * (-sin(th) / -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 <= 1.2d-150) then
        tmp = ky * (-sin(th) / -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 <= 1.2e-150) {
		tmp = ky * (-Math.sin(th) / -kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.2e-150:
		tmp = ky * (-math.sin(th) / -kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.2e-150)
		tmp = Float64(ky * Float64(Float64(-sin(th)) / Float64(-kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.2e-150)
		tmp = ky * (-sin(th) / -kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.2e-150], N[(ky * N[((-N[Sin[th], $MachinePrecision]) / (-kx)), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

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Alternative 11: 26.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 9.5 \cdot 10^{-150}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 9.5e-150) (* (sin th) (/ ky kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 9.5e-150) {
		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 <= 9.5d-150) 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 <= 9.5e-150) {
		tmp = Math.sin(th) * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 9.5e-150:
		tmp = math.sin(th) * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 9.5e-150)
		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 <= 9.5e-150)
		tmp = sin(th) * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 9.5e-150], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

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Alternative 12: 26.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2.3 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 2.3e-149) (/ (sin th) (/ kx ky)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.3e-149) {
		tmp = sin(th) / (kx / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 2.3d-149) then
        tmp = sin(th) / (kx / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.3e-149) {
		tmp = Math.sin(th) / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 2.3e-149:
		tmp = math.sin(th) / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 2.3e-149)
		tmp = Float64(sin(th) / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 2.3e-149)
		tmp = sin(th) / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 2.3e-149], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

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Alternative 13: 24.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 4.6 \cdot 10^{-160}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 4.6e-160) (* th (/ ky kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.6e-160) {
		tmp = 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 <= 4.6d-160) then
        tmp = 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 <= 4.6e-160) {
		tmp = th * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 4.6e-160:
		tmp = th * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 4.6e-160)
		tmp = Float64(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 <= 4.6e-160)
		tmp = th * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 4.6e-160], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

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Alternative 14: 13.4% accurate, 141.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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Reproduce

herbie shell --seed 2024010 
(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)))