Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.7%
Time: 11.8s
Alternatives: 10
Speedup: 1.4×

Specification

?
\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]
(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))
double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

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

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);
}

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)

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

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

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]

\begin{array}{l}

\\
\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 10 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 93.8% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]
(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))
double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

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

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);
}

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)

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

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

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]

\begin{array}{l}

\\
\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th
\end{array}

Alternative 1: 99.7% accurate, 1.4× speedup?

\[\begin{array}{l} \\ \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \end{array} \]
(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (hypot (sin ky) (sin kx))) (sin th)))
double code(double kx, double ky, double th) {
	return (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
}

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);
}

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

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

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

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]

\begin{array}{l}

\\
\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th
\end{array}
Derivation
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Alternative 2: 99.6% accurate, 1.4× speedup?

\[\begin{array}{l} \\ \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \end{array} \]
(FPCore (kx ky th)
 :precision binary64
 (* (sin ky) (/ (sin th) (hypot (sin ky) (sin kx)))))
double code(double kx, double ky, double th) {
	return sin(ky) * (sin(th) / hypot(sin(ky), sin(kx)));
}

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)));
}

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

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

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

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]

\begin{array}{l}

\\
\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}
\end{array}
Derivation
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Alternative 3: 33.2% accurate, 1.7× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.29:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]
(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.29) (* (sin th) (fabs (/ (sin ky) (sin kx)))) (sin th)))
double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.29) {
		tmp = sin(th) * fabs((sin(ky) / sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

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 <= 0.29d0) then
        tmp = sin(th) * abs((sin(ky) / sin(kx)))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.29) {
		tmp = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.29:
		tmp = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.29)
		tmp = Float64(sin(th) * abs(Float64(sin(ky) / sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.29)
		tmp = sin(th) * abs((sin(ky) / sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

code[kx_, ky_, th_] := If[LessEqual[ky, 0.29], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;ky \leq 0.29:\\
\;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\

\mathbf{else}:\\
\;\;\;\;\sin th\\


\end{array}
\end{array}
Derivation
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Alternative 4: 40.6% accurate, 2.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sin th}{ky \cdot \frac{0.5}{kx} + \frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]
(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-12)
   (/ (sin th) (+ (* ky (/ 0.5 kx)) (/ (sin kx) ky)))
   (sin th)))
double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-12) {
		tmp = sin(th) / ((ky * (0.5 / kx)) + (sin(kx) / ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 2d-12) then
        tmp = sin(th) / ((ky * (0.5d0 / kx)) + (sin(kx) / ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 2e-12) {
		tmp = Math.sin(th) / ((ky * (0.5 / kx)) + (Math.sin(kx) / ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 2e-12:
		tmp = math.sin(th) / ((ky * (0.5 / kx)) + (math.sin(kx) / ky))
	else:
		tmp = math.sin(th)
	return tmp

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-12)
		tmp = Float64(sin(th) / Float64(Float64(ky * Float64(0.5 / kx)) + Float64(sin(kx) / ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 2e-12)
		tmp = sin(th) / ((ky * (0.5 / kx)) + (sin(kx) / ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-12], N[(N[Sin[th], $MachinePrecision] / N[(N[(ky * N[(0.5 / kx), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\sin ky \leq 2 \cdot 10^{-12}:\\
\;\;\;\;\frac{\sin th}{ky \cdot \frac{0.5}{kx} + \frac{\sin kx}{ky}}\\

\mathbf{else}:\\
\;\;\;\;\sin th\\


\end{array}
\end{array}
Derivation
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Alternative 5: 32.0% accurate, 2.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.29:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]
(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.29) (* (sin th) (fabs (/ ky (sin kx)))) (sin th)))
double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.29) {
		tmp = sin(th) * fabs((ky / sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

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 <= 0.29d0) then
        tmp = sin(th) * abs((ky / sin(kx)))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.29) {
		tmp = Math.sin(th) * Math.abs((ky / Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.29:
		tmp = math.sin(th) * math.fabs((ky / math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.29)
		tmp = Float64(sin(th) * abs(Float64(ky / sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.29)
		tmp = sin(th) * abs((ky / sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

code[kx_, ky_, th_] := If[LessEqual[ky, 0.29], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;ky \leq 0.29:\\
\;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\

\mathbf{else}:\\
\;\;\;\;\sin th\\


\end{array}
\end{array}
Derivation
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Alternative 6: 32.3% accurate, 3.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 190000:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]
(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 190000.0) (* (sin th) (/ ky (sin kx))) (sin th)))
double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 190000.0) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

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 <= 190000.0d0) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 190000.0) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

def code(kx, ky, th):
	tmp = 0
	if ky <= 190000.0:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 190000.0)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 190000.0)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

code[kx_, ky_, th_] := If[LessEqual[ky, 190000.0], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;ky \leq 190000:\\
\;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\

\mathbf{else}:\\
\;\;\;\;\sin th\\


\end{array}
\end{array}
Derivation
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Alternative 7: 32.3% accurate, 3.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 190000:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]
(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 190000.0) (* ky (/ (sin th) (sin kx))) (sin th)))
double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 190000.0) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

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 <= 190000.0d0) then
        tmp = ky * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 190000.0) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

def code(kx, ky, th):
	tmp = 0
	if ky <= 190000.0:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 190000.0)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 190000.0)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

code[kx_, ky_, th_] := If[LessEqual[ky, 190000.0], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;ky \leq 190000:\\
\;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\

\mathbf{else}:\\
\;\;\;\;\sin th\\


\end{array}
\end{array}
Derivation
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Alternative 8: 26.2% accurate, 6.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 6.4 \cdot 10^{-91}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]
(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 6.4e-91) (* (sin th) (/ ky kx)) (sin th)))
double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 6.4e-91) {
		tmp = sin(th) * (ky / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

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 <= 6.4d-91) then
        tmp = sin(th) * (ky / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 6.4e-91) {
		tmp = Math.sin(th) * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

def code(kx, ky, th):
	tmp = 0
	if ky <= 6.4e-91:
		tmp = math.sin(th) * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 6.4e-91)
		tmp = Float64(sin(th) * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 6.4e-91)
		tmp = sin(th) * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

code[kx_, ky_, th_] := If[LessEqual[ky, 6.4e-91], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;ky \leq 6.4 \cdot 10^{-91}:\\
\;\;\;\;\sin th \cdot \frac{ky}{kx}\\

\mathbf{else}:\\
\;\;\;\;\sin th\\


\end{array}
\end{array}
Derivation
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  1. Add Preprocessing

Alternative 9: 22.8% accurate, 6.9× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.35 \cdot 10^{-127}:\\ \;\;\;\;\frac{ky \cdot th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]
(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.35e-127) (/ (* ky th) kx) (sin th)))
double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.35e-127) {
		tmp = (ky * th) / kx;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

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 <= 1.35d-127) then
        tmp = (ky * th) / kx
    else
        tmp = sin(th)
    end if
    code = tmp
end function

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.35e-127) {
		tmp = (ky * th) / kx;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.35e-127:
		tmp = (ky * th) / kx
	else:
		tmp = math.sin(th)
	return tmp

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.35e-127)
		tmp = Float64(Float64(ky * th) / kx);
	else
		tmp = sin(th);
	end
	return tmp
end

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.35e-127)
		tmp = (ky * th) / kx;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

code[kx_, ky_, th_] := If[LessEqual[ky, 1.35e-127], N[(N[(ky * th), $MachinePrecision] / kx), $MachinePrecision], N[Sin[th], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;ky \leq 1.35 \cdot 10^{-127}:\\
\;\;\;\;\frac{ky \cdot th}{kx}\\

\mathbf{else}:\\
\;\;\;\;\sin th\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 10: 12.3% accurate, 141.8× speedup?

\[\begin{array}{l} \\ \frac{ky \cdot th}{kx} \end{array} \]
(FPCore (kx ky th) :precision binary64 (/ (* ky th) kx))
double code(double kx, double ky, double th) {
	return (ky * th) / kx;
}

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

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

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

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

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

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

\begin{array}{l}

\\
\frac{ky \cdot th}{kx}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2023340 
(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)))