Octave 3.8, jcobi/1

Percentage Accurate: 74.4% → 99.9%
Time: 9.6s
Alternatives: 13
Speedup: 1.2×

Specification

?
\[\alpha > -1 \land \beta > -1\]
\[\begin{array}{l} \\ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))
double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

public static double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

def code(alpha, beta):
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
end

function tmp = code(alpha, beta)
	tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
end

code[alpha_, beta_] := N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

\begin{array}{l}

\\
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
\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 13 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: 74.4% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))
double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

public static double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

def code(alpha, beta):
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
end

function tmp = code(alpha, beta)
	tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
end

code[alpha_, beta_] := N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

\begin{array}{l}

\\
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
\end{array}

Alternative 1: 99.9% accurate, 0.1× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2 - \beta\right) - \beta\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot t_0 - \frac{t_0}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (- -2.0 beta) beta)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999998)
     (/ (- (* (/ (+ beta 2.0) (pow alpha 2.0)) t_0) (/ t_0 alpha)) 2.0)
     (/ (fma (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))) 1.0) 2.0))))
double code(double alpha, double beta) {
	double t_0 = (-2.0 - beta) - beta;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998) {
		tmp = ((((beta + 2.0) / pow(alpha, 2.0)) * t_0) - (t_0 / alpha)) / 2.0;
	} else {
		tmp = fma((beta - alpha), (1.0 / (beta + (alpha + 2.0))), 1.0) / 2.0;
	}
	return tmp;
}

function code(alpha, beta)
	t_0 = Float64(Float64(-2.0 - beta) - beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999998)
		tmp = Float64(Float64(Float64(Float64(Float64(beta + 2.0) / (alpha ^ 2.0)) * t_0) - Float64(t_0 / alpha)) / 2.0);
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), 1.0) / 2.0);
	end
	return tmp
end

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999998], N[(N[(N[(N[(N[(beta + 2.0), $MachinePrecision] / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \left(-2 - \beta\right) - \beta\\
\mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\
\;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot t_0 - \frac{t_0}{\alpha}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\


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

Alternative 2: 99.4% accurate, 0.1× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -1:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -1.0)
   (/ (+ (/ beta alpha) (/ (- beta -2.0) alpha)) 2.0)
   (/ (fma (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))) 1.0) 2.0)))
double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0) {
		tmp = ((beta / alpha) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = fma((beta - alpha), (1.0 / (beta + (alpha + 2.0))), 1.0) / 2.0;
	}
	return tmp;
}

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -1.0)
		tmp = Float64(Float64(Float64(beta / alpha) + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), 1.0) / 2.0);
	end
	return tmp
end

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(beta / alpha), $MachinePrecision] + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -1:\\
\;\;\;\;\frac{\frac{\beta}{\alpha} + \frac{\beta - -2}{\alpha}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\


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

Alternative 3: 99.4% accurate, 0.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -1.0)
     (/ (+ (/ beta alpha) (/ (- beta -2.0) alpha)) 2.0)
     (/ (+ t_0 1.0) 2.0))))
double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = ((beta / alpha) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-1.0d0)) then
        tmp = ((beta / alpha) + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = ((beta / alpha) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -1.0:
		tmp = ((beta / alpha) + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = Float64(Float64(Float64(beta / alpha) + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -1.0)
		tmp = ((beta / alpha) + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[(N[(N[(beta / alpha), $MachinePrecision] + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\
\mathbf{if}\;t_0 \leq -1:\\
\;\;\;\;\frac{\frac{\beta}{\alpha} + \frac{\beta - -2}{\alpha}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{t_0 + 1}{2}\\


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

Alternative 4: 93.6% accurate, 0.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{-19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 23000000:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3900000000:\\ \;\;\;\;\frac{2 - 2 \cdot \frac{\alpha}{\beta}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \frac{\beta - -2}{\alpha}}{2}\\ \end{array} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3.7e-19)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (<= alpha 23000000.0)
     (/ (- 1.0 (/ alpha (+ alpha 2.0))) 2.0)
     (if (<= alpha 3900000000.0)
       (/ (- 2.0 (* 2.0 (/ alpha beta))) 2.0)
       (/ (+ (/ beta alpha) (/ (- beta -2.0) alpha)) 2.0)))))
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.7e-19) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 23000000.0) {
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	} else if (alpha <= 3900000000.0) {
		tmp = (2.0 - (2.0 * (alpha / beta))) / 2.0;
	} else {
		tmp = ((beta / alpha) + ((beta - -2.0) / alpha)) / 2.0;
	}
	return tmp;
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 3.7d-19) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if (alpha <= 23000000.0d0) then
        tmp = (1.0d0 - (alpha / (alpha + 2.0d0))) / 2.0d0
    else if (alpha <= 3900000000.0d0) then
        tmp = (2.0d0 - (2.0d0 * (alpha / beta))) / 2.0d0
    else
        tmp = ((beta / alpha) + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    end if
    code = tmp
end function

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.7e-19) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 23000000.0) {
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	} else if (alpha <= 3900000000.0) {
		tmp = (2.0 - (2.0 * (alpha / beta))) / 2.0;
	} else {
		tmp = ((beta / alpha) + ((beta - -2.0) / alpha)) / 2.0;
	}
	return tmp;
}

def code(alpha, beta):
	tmp = 0
	if alpha <= 3.7e-19:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif alpha <= 23000000.0:
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0
	elif alpha <= 3900000000.0:
		tmp = (2.0 - (2.0 * (alpha / beta))) / 2.0
	else:
		tmp = ((beta / alpha) + ((beta - -2.0) / alpha)) / 2.0
	return tmp

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3.7e-19)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif (alpha <= 23000000.0)
		tmp = Float64(Float64(1.0 - Float64(alpha / Float64(alpha + 2.0))) / 2.0);
	elseif (alpha <= 3900000000.0)
		tmp = Float64(Float64(2.0 - Float64(2.0 * Float64(alpha / beta))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / alpha) + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	end
	return tmp
end

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 3.7e-19)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif (alpha <= 23000000.0)
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	elseif (alpha <= 3900000000.0)
		tmp = (2.0 - (2.0 * (alpha / beta))) / 2.0;
	else
		tmp = ((beta / alpha) + ((beta - -2.0) / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

code[alpha_, beta_] := If[LessEqual[alpha, 3.7e-19], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 23000000.0], N[(N[(1.0 - N[(alpha / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 3900000000.0], N[(N[(2.0 - N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / alpha), $MachinePrecision] + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\alpha \leq 3.7 \cdot 10^{-19}:\\
\;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\

\mathbf{elif}\;\alpha \leq 23000000:\\
\;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\

\mathbf{elif}\;\alpha \leq 3900000000:\\
\;\;\;\;\frac{2 - 2 \cdot \frac{\alpha}{\beta}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\alpha} + \frac{\beta - -2}{\alpha}}{2}\\


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

Alternative 5: 88.8% accurate, 0.9× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 25000000:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 13600000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))
   (if (<= alpha 3.7e-19)
     t_0
     (if (<= alpha 25000000.0)
       (/ (- 1.0 (/ alpha (+ alpha 2.0))) 2.0)
       (if (<= alpha 13600000000000.0) t_0 (/ (/ (+ beta 2.0) alpha) 2.0))))))
double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 3.7e-19) {
		tmp = t_0;
	} else if (alpha <= 25000000.0) {
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	} else if (alpha <= 13600000000000.0) {
		tmp = t_0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    if (alpha <= 3.7d-19) then
        tmp = t_0
    else if (alpha <= 25000000.0d0) then
        tmp = (1.0d0 - (alpha / (alpha + 2.0d0))) / 2.0d0
    else if (alpha <= 13600000000000.0d0) then
        tmp = t_0
    else
        tmp = ((beta + 2.0d0) / alpha) / 2.0d0
    end if
    code = tmp
end function

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 3.7e-19) {
		tmp = t_0;
	} else if (alpha <= 25000000.0) {
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	} else if (alpha <= 13600000000000.0) {
		tmp = t_0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

def code(alpha, beta):
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0
	tmp = 0
	if alpha <= 3.7e-19:
		tmp = t_0
	elif alpha <= 25000000.0:
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0
	elif alpha <= 13600000000000.0:
		tmp = t_0
	else:
		tmp = ((beta + 2.0) / alpha) / 2.0
	return tmp

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0)
	tmp = 0.0
	if (alpha <= 3.7e-19)
		tmp = t_0;
	elseif (alpha <= 25000000.0)
		tmp = Float64(Float64(1.0 - Float64(alpha / Float64(alpha + 2.0))) / 2.0);
	elseif (alpha <= 13600000000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(beta + 2.0) / alpha) / 2.0);
	end
	return tmp
end

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	tmp = 0.0;
	if (alpha <= 3.7e-19)
		tmp = t_0;
	elseif (alpha <= 25000000.0)
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	elseif (alpha <= 13600000000000.0)
		tmp = t_0;
	else
		tmp = ((beta + 2.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 3.7e-19], t$95$0, If[LessEqual[alpha, 25000000.0], N[(N[(1.0 - N[(alpha / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 13600000000000.0], t$95$0, N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\
\mathbf{if}\;\alpha \leq 3.7 \cdot 10^{-19}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;\alpha \leq 25000000:\\
\;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\

\mathbf{elif}\;\alpha \leq 13600000000000:\\
\;\;\;\;t_0\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\


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

Alternative 6: 88.7% accurate, 0.9× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{-19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 25000000:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3800000000:\\ \;\;\;\;\frac{2 - 2 \cdot \frac{\alpha}{\beta}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3.7e-19)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (<= alpha 25000000.0)
     (/ (- 1.0 (/ alpha (+ alpha 2.0))) 2.0)
     (if (<= alpha 3800000000.0)
       (/ (- 2.0 (* 2.0 (/ alpha beta))) 2.0)
       (/ (/ (+ beta 2.0) alpha) 2.0)))))
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.7e-19) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 25000000.0) {
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	} else if (alpha <= 3800000000.0) {
		tmp = (2.0 - (2.0 * (alpha / beta))) / 2.0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 3.7d-19) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if (alpha <= 25000000.0d0) then
        tmp = (1.0d0 - (alpha / (alpha + 2.0d0))) / 2.0d0
    else if (alpha <= 3800000000.0d0) then
        tmp = (2.0d0 - (2.0d0 * (alpha / beta))) / 2.0d0
    else
        tmp = ((beta + 2.0d0) / alpha) / 2.0d0
    end if
    code = tmp
end function

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.7e-19) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 25000000.0) {
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	} else if (alpha <= 3800000000.0) {
		tmp = (2.0 - (2.0 * (alpha / beta))) / 2.0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

def code(alpha, beta):
	tmp = 0
	if alpha <= 3.7e-19:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif alpha <= 25000000.0:
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0
	elif alpha <= 3800000000.0:
		tmp = (2.0 - (2.0 * (alpha / beta))) / 2.0
	else:
		tmp = ((beta + 2.0) / alpha) / 2.0
	return tmp

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3.7e-19)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif (alpha <= 25000000.0)
		tmp = Float64(Float64(1.0 - Float64(alpha / Float64(alpha + 2.0))) / 2.0);
	elseif (alpha <= 3800000000.0)
		tmp = Float64(Float64(2.0 - Float64(2.0 * Float64(alpha / beta))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + 2.0) / alpha) / 2.0);
	end
	return tmp
end

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 3.7e-19)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif (alpha <= 25000000.0)
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	elseif (alpha <= 3800000000.0)
		tmp = (2.0 - (2.0 * (alpha / beta))) / 2.0;
	else
		tmp = ((beta + 2.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

code[alpha_, beta_] := If[LessEqual[alpha, 3.7e-19], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 25000000.0], N[(N[(1.0 - N[(alpha / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 3800000000.0], N[(N[(2.0 - N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\alpha \leq 3.7 \cdot 10^{-19}:\\
\;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\

\mathbf{elif}\;\alpha \leq 25000000:\\
\;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\

\mathbf{elif}\;\alpha \leq 3800000000:\\
\;\;\;\;\frac{2 - 2 \cdot \frac{\alpha}{\beta}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\


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

Alternative 7: 93.6% accurate, 0.9× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{-19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 23000000:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 30000000000:\\ \;\;\;\;\frac{2 - 2 \cdot \frac{\alpha}{\beta}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3.7e-19)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (<= alpha 23000000.0)
     (/ (- 1.0 (/ alpha (+ alpha 2.0))) 2.0)
     (if (<= alpha 30000000000.0)
       (/ (- 2.0 (* 2.0 (/ alpha beta))) 2.0)
       (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))))
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.7e-19) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 23000000.0) {
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	} else if (alpha <= 30000000000.0) {
		tmp = (2.0 - (2.0 * (alpha / beta))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 3.7d-19) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if (alpha <= 23000000.0d0) then
        tmp = (1.0d0 - (alpha / (alpha + 2.0d0))) / 2.0d0
    else if (alpha <= 30000000000.0d0) then
        tmp = (2.0d0 - (2.0d0 * (alpha / beta))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.7e-19) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 23000000.0) {
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	} else if (alpha <= 30000000000.0) {
		tmp = (2.0 - (2.0 * (alpha / beta))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

def code(alpha, beta):
	tmp = 0
	if alpha <= 3.7e-19:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif alpha <= 23000000.0:
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0
	elif alpha <= 30000000000.0:
		tmp = (2.0 - (2.0 * (alpha / beta))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3.7e-19)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif (alpha <= 23000000.0)
		tmp = Float64(Float64(1.0 - Float64(alpha / Float64(alpha + 2.0))) / 2.0);
	elseif (alpha <= 30000000000.0)
		tmp = Float64(Float64(2.0 - Float64(2.0 * Float64(alpha / beta))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 3.7e-19)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif (alpha <= 23000000.0)
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	elseif (alpha <= 30000000000.0)
		tmp = (2.0 - (2.0 * (alpha / beta))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

code[alpha_, beta_] := If[LessEqual[alpha, 3.7e-19], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 23000000.0], N[(N[(1.0 - N[(alpha / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 30000000000.0], N[(N[(2.0 - N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\alpha \leq 3.7 \cdot 10^{-19}:\\
\;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\

\mathbf{elif}\;\alpha \leq 23000000:\\
\;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\

\mathbf{elif}\;\alpha \leq 30000000000:\\
\;\;\;\;\frac{2 - 2 \cdot \frac{\alpha}{\beta}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\


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

Alternative 8: 69.8% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -6.5 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -1.85 \cdot 10^{-212}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)))
   (if (<= beta -6.5e-156)
     t_0
     (if (<= beta -1.85e-212)
       (/ (/ 2.0 alpha) 2.0)
       (if (<= beta 2.0) t_0 1.0)))))
double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= -6.5e-156) {
		tmp = t_0;
	} else if (beta <= -1.85e-212) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    if (beta <= (-6.5d-156)) then
        tmp = t_0
    else if (beta <= (-1.85d-212)) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= -6.5e-156) {
		tmp = t_0;
	} else if (beta <= -1.85e-212) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

def code(alpha, beta):
	t_0 = (1.0 + (beta * 0.5)) / 2.0
	tmp = 0
	if beta <= -6.5e-156:
		tmp = t_0
	elif beta <= -1.85e-212:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0)
	tmp = 0.0
	if (beta <= -6.5e-156)
		tmp = t_0;
	elseif (beta <= -1.85e-212)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta * 0.5)) / 2.0;
	tmp = 0.0;
	if (beta <= -6.5e-156)
		tmp = t_0;
	elseif (beta <= -1.85e-212)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, -6.5e-156], t$95$0, If[LessEqual[beta, -1.85e-212], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 2.0], t$95$0, 1.0]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{1 + \beta \cdot 0.5}{2}\\
\mathbf{if}\;\beta \leq -6.5 \cdot 10^{-156}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;\beta \leq -1.85 \cdot 10^{-212}:\\
\;\;\;\;\frac{\frac{2}{\alpha}}{2}\\

\mathbf{elif}\;\beta \leq 2:\\
\;\;\;\;t_0\\

\mathbf{else}:\\
\;\;\;\;1\\


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

Alternative 9: 70.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -8.2 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -1.45 \cdot 10^{-212}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)))
   (if (<= beta -8.2e-156)
     t_0
     (if (<= beta -1.45e-212)
       (/ (/ 2.0 alpha) 2.0)
       (if (<= beta 2.0) t_0 (/ (- 2.0 (/ 2.0 beta)) 2.0))))))
double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= -8.2e-156) {
		tmp = t_0;
	} else if (beta <= -1.45e-212) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    if (beta <= (-8.2d-156)) then
        tmp = t_0
    else if (beta <= (-1.45d-212)) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = (2.0d0 - (2.0d0 / beta)) / 2.0d0
    end if
    code = tmp
end function

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= -8.2e-156) {
		tmp = t_0;
	} else if (beta <= -1.45e-212) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

def code(alpha, beta):
	t_0 = (1.0 + (beta * 0.5)) / 2.0
	tmp = 0
	if beta <= -8.2e-156:
		tmp = t_0
	elif beta <= -1.45e-212:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = (2.0 - (2.0 / beta)) / 2.0
	return tmp

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0)
	tmp = 0.0
	if (beta <= -8.2e-156)
		tmp = t_0;
	elseif (beta <= -1.45e-212)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 - Float64(2.0 / beta)) / 2.0);
	end
	return tmp
end

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta * 0.5)) / 2.0;
	tmp = 0.0;
	if (beta <= -8.2e-156)
		tmp = t_0;
	elseif (beta <= -1.45e-212)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	end
	tmp_2 = tmp;
end

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, -8.2e-156], t$95$0, If[LessEqual[beta, -1.45e-212], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 2.0], t$95$0, N[(N[(2.0 - N[(2.0 / beta), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{1 + \beta \cdot 0.5}{2}\\
\mathbf{if}\;\beta \leq -8.2 \cdot 10^{-156}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;\beta \leq -1.45 \cdot 10^{-212}:\\
\;\;\;\;\frac{\frac{2}{\alpha}}{2}\\

\mathbf{elif}\;\beta \leq 2:\\
\;\;\;\;t_0\\

\mathbf{else}:\\
\;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\


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

Alternative 10: 88.8% accurate, 1.2× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 44000000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 44000000000000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta 2.0) alpha) 2.0)))
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 44000000000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 44000000000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + 2.0d0) / alpha) / 2.0d0
    end if
    code = tmp
end function

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 44000000000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

def code(alpha, beta):
	tmp = 0
	if alpha <= 44000000000000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + 2.0) / alpha) / 2.0
	return tmp

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 44000000000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + 2.0) / alpha) / 2.0);
	end
	return tmp
end

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 44000000000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + 2.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

code[alpha_, beta_] := If[LessEqual[alpha, 44000000000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\alpha \leq 44000000000000:\\
\;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\


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

Alternative 11: 52.4% accurate, 1.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.6 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 5.6e+16) 1.0 (/ (/ 2.0 alpha) 2.0)))
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5.6e+16) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 5.6d+16) then
        tmp = 1.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5.6e+16) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

def code(alpha, beta):
	tmp = 0
	if alpha <= 5.6e+16:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 5.6e+16)
		tmp = 1.0;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 5.6e+16)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

code[alpha_, beta_] := If[LessEqual[alpha, 5.6e+16], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\alpha \leq 5.6 \cdot 10^{+16}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{\alpha}}{2}\\


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

Alternative 12: 40.1% accurate, 2.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7 \cdot 10^{+143}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \end{array} \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7e+143) 1.0 (/ beta alpha)))
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7e+143) {
		tmp = 1.0;
	} else {
		tmp = beta / alpha;
	}
	return tmp;
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 7d+143) then
        tmp = 1.0d0
    else
        tmp = beta / alpha
    end if
    code = tmp
end function

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7e+143) {
		tmp = 1.0;
	} else {
		tmp = beta / alpha;
	}
	return tmp;
}

def code(alpha, beta):
	tmp = 0
	if alpha <= 7e+143:
		tmp = 1.0
	else:
		tmp = beta / alpha
	return tmp

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 7e+143)
		tmp = 1.0;
	else
		tmp = Float64(beta / alpha);
	end
	return tmp
end

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 7e+143)
		tmp = 1.0;
	else
		tmp = beta / alpha;
	end
	tmp_2 = tmp;
end

code[alpha_, beta_] := If[LessEqual[alpha, 7e+143], 1.0, N[(beta / alpha), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\alpha \leq 7 \cdot 10^{+143}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{\beta}{\alpha}\\


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

Alternative 13: 36.9% accurate, 13.0× speedup?

\[\begin{array}{l} \\ 1 \end{array} \]
(FPCore (alpha beta) :precision binary64 1.0)
double code(double alpha, double beta) {
	return 1.0;
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 1.0d0
end function

public static double code(double alpha, double beta) {
	return 1.0;
}

def code(alpha, beta):
	return 1.0

function code(alpha, beta)
	return 1.0
end

function tmp = code(alpha, beta)
	tmp = 1.0;
end

code[alpha_, beta_] := 1.0

\begin{array}{l}

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

Reproduce

?
herbie shell --seed 2023342 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))