Octave 3.8, jcobi/1

Percentage Accurate: 75.0% → 99.6%

Time: 11.2s

Alternatives: 10

Speedup: 0.9×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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]

Initial Program: 75.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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]

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\left(-2 - \beta\right) \cdot \frac{\beta + \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{t\_0}\right) - \frac{\alpha}{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.5)
     (/
      (+
       (* (- -2.0 beta) (/ (+ beta (+ beta 2.0)) (pow alpha 2.0)))
       (/ (+ beta (- beta -2.0)) alpha))
      2.0)
     (/ (- (+ 1.0 (/ beta t_0)) (/ alpha t_0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.5) {
		tmp = (((-2.0 - beta) * ((beta + (beta + 2.0)) / pow(alpha, 2.0))) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_0)) - (alpha / t_0)) / 2.0;
	}
	return tmp;
}

Fortran

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 + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.5d0)) then
        tmp = ((((-2.0d0) - beta) * ((beta + (beta + 2.0d0)) / (alpha ** 2.0d0))) + ((beta + (beta - (-2.0d0))) / alpha)) / 2.0d0
    else
        tmp = ((1.0d0 + (beta / t_0)) - (alpha / t_0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.5) {
		tmp = (((-2.0 - beta) * ((beta + (beta + 2.0)) / Math.pow(alpha, 2.0))) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_0)) - (alpha / t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.5:
		tmp = (((-2.0 - beta) * ((beta + (beta + 2.0)) / math.pow(alpha, 2.0))) + ((beta + (beta - -2.0)) / alpha)) / 2.0
	else:
		tmp = ((1.0 + (beta / t_0)) - (alpha / t_0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.5)
		tmp = Float64(Float64(Float64(Float64(-2.0 - beta) * Float64(Float64(beta + Float64(beta + 2.0)) / (alpha ^ 2.0))) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(beta / t_0)) - Float64(alpha / t_0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.5)
		tmp = (((-2.0 - beta) * ((beta + (beta + 2.0)) / (alpha ^ 2.0))) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	else
		tmp = ((1.0 + (beta / t_0)) - (alpha / t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.5

   1. Initial program 6.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around -inf 98.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\beta \cdot \left(2 + \beta\right) + {\left(2 + \beta\right)}^{2}}{{\alpha}^{2}} + -1 \cdot \frac{-1 \cdot \beta - \left(2 + \beta\right)}{\alpha}}}{2} \]
   4. Step-by-step derivation

      1. mul-1-neg 98.0%

         \[\leadsto \frac{-1 \cdot \frac{\beta \cdot \left(2 + \beta\right) + {\left(2 + \beta\right)}^{2}}{{\alpha}^{2}} + \color{blue}{\left(-\frac{-1 \cdot \beta - \left(2 + \beta\right)}{\alpha}\right)}}{2} \]

      2. unsub-neg 98.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\beta \cdot \left(2 + \beta\right) + {\left(2 + \beta\right)}^{2}}{{\alpha}^{2}} - \frac{-1 \cdot \beta - \left(2 + \beta\right)}{\alpha}}}{2} \]

   5. Simplified 99.6%

      \[\leadsto \frac{\color{blue}{\left(-2 - \beta\right) \cdot \frac{\beta + \left(2 + \beta\right)}{{\alpha}^{2}} - \frac{\left(-2 - \beta\right) - \beta}{\alpha}}}{2} \]

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 100.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{\color{blue}{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}}{2} \]

      2. div-sub 100.0%

         \[\leadsto \frac{1 + \color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}}{2} \]

      3. associate-+r- 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}\right) - \frac{\alpha}{\left(\alpha + \beta\right) + 2}}}{2} \]

      4. +-commutative 100.0%

         \[\leadsto \frac{\left(1 + \frac{\beta}{\color{blue}{\left(\beta + \alpha\right)} + 2}\right) - \frac{\alpha}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+r+ 100.0%

         \[\leadsto \frac{\left(1 + \frac{\beta}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right) - \frac{\alpha}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. +-commutative 100.0%

         \[\leadsto \frac{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) - \frac{\alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2}}{2} \]

      7. associate-+r+ 100.0%

         \[\leadsto \frac{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) - \frac{\alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}}{2} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) - \frac{\alpha}{\beta + \left(\alpha + 2\right)}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\left(-2 - \beta\right) \cdot \frac{\beta + \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) - \frac{\alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{t\_0}\right) - \frac{\alpha}{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999996)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (- (+ 1.0 (/ beta t_0)) (/ alpha t_0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_0)) - (alpha / t_0)) / 2.0;
	}
	return tmp;
}

Fortran

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 + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999996d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((1.0d0 + (beta / t_0)) - (alpha / t_0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_0)) - (alpha / t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = ((1.0 + (beta / t_0)) - (alpha / t_0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999996)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(beta / t_0)) - Float64(alpha / t_0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = ((1.0 + (beta / t_0)) - (alpha / t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999996], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999995999999999996

   1. Initial program 5.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf 99.9%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]

   if -0.999995999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \frac{\color{blue}{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}}{2} \]

      2. div-sub 99.9%

         \[\leadsto \frac{1 + \color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}}{2} \]

      3. associate-+r- 99.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}\right) - \frac{\alpha}{\left(\alpha + \beta\right) + 2}}}{2} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\left(1 + \frac{\beta}{\color{blue}{\left(\beta + \alpha\right)} + 2}\right) - \frac{\alpha}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+r+ 99.9%

         \[\leadsto \frac{\left(1 + \frac{\beta}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right) - \frac{\alpha}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. +-commutative 99.9%

         \[\leadsto \frac{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) - \frac{\alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2}}{2} \]

      7. associate-+r+ 99.9%

         \[\leadsto \frac{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) - \frac{\alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}}{2} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) - \frac{\alpha}{\beta + \left(\alpha + 2\right)}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) - \frac{\alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -4.8 \cdot 10^{-141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq -7.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 17500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)))
   (if (<= beta -4.8e-141)
     t_0
     (if (<= beta -7.2e-176)
       (/ (/ 2.0 alpha) 2.0)
       (if (<= beta 17500.0)
         t_0
         (if (<= beta 7.2e+47) (/ (* 2.0 (/ beta alpha)) 2.0) 1.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= -4.8e-141) {
		tmp = t_0;
	} else if (beta <= -7.2e-176) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 17500.0) {
		tmp = t_0;
	} else if (beta <= 7.2e+47) {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

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 <= (-4.8d-141)) then
        tmp = t_0
    else if (beta <= (-7.2d-176)) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 17500.0d0) then
        tmp = t_0
    else if (beta <= 7.2d+47) then
        tmp = (2.0d0 * (beta / alpha)) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= -4.8e-141) {
		tmp = t_0;
	} else if (beta <= -7.2e-176) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 17500.0) {
		tmp = t_0;
	} else if (beta <= 7.2e+47) {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (beta * 0.5)) / 2.0
	tmp = 0
	if beta <= -4.8e-141:
		tmp = t_0
	elif beta <= -7.2e-176:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 17500.0:
		tmp = t_0
	elif beta <= 7.2e+47:
		tmp = (2.0 * (beta / alpha)) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0)
	tmp = 0.0
	if (beta <= -4.8e-141)
		tmp = t_0;
	elseif (beta <= -7.2e-176)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 17500.0)
		tmp = t_0;
	elseif (beta <= 7.2e+47)
		tmp = Float64(Float64(2.0 * Float64(beta / alpha)) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta * 0.5)) / 2.0;
	tmp = 0.0;
	if (beta <= -4.8e-141)
		tmp = t_0;
	elseif (beta <= -7.2e-176)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 17500.0)
		tmp = t_0;
	elseif (beta <= 7.2e+47)
		tmp = (2.0 * (beta / alpha)) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, -4.8e-141], t$95$0, If[LessEqual[beta, -7.2e-176], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 17500.0], t$95$0, If[LessEqual[beta, 7.2e+47], N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < -4.8000000000000002e-141 or -7.2000000000000005e-176 < beta < 17500

   1. Initial program 74.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0 72.8%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]

   4. Taylor expanded in beta around 0 72.1%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \beta} + 1}{2} \]
   5. Step-by-step derivation

      1. *-commutative 72.1%

         \[\leadsto \frac{\color{blue}{\beta \cdot 0.5} + 1}{2} \]

   6. Simplified 72.1%

      \[\leadsto \frac{\color{blue}{\beta \cdot 0.5} + 1}{2} \]

   if -4.8000000000000002e-141 < beta < -7.2000000000000005e-176

   1. Initial program 31.2%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around 0 31.2%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{2 + \alpha}}}{2} \]
   4. Step-by-step derivation

      1. +-commutative 31.2%

         \[\leadsto \frac{1 - \frac{\alpha}{\color{blue}{\alpha + 2}}}{2} \]

   5. Simplified 31.2%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{\alpha + 2}}}{2} \]

   6. Taylor expanded in alpha around inf 73.4%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha}}}{2} \]

   if 17500 < beta < 7.20000000000000015e47

   1. Initial program 26.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf 79.7%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]

   4. Taylor expanded in beta around inf 76.7%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{\beta}{\alpha}}}{2} \]

   if 7.20000000000000015e47 < beta

   1. Initial program 91.4%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 91.4%

         \[\leadsto \frac{\color{blue}{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}}{2} \]

      2. sub-neg 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{\beta + \left(-\alpha\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      3. +-commutative 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(-\alpha\right) + \beta}}{\left(\alpha + \beta\right) + 2}}{2} \]

      4. neg-sub0 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{-\left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      7. distribute-frac-neg 91.4%

         \[\leadsto \frac{1 + \color{blue}{\left(-\frac{\alpha - \beta}{\left(\alpha + \beta\right) + 2}\right)}}{2} \]

      8. +-commutative 91.4%

         \[\leadsto \frac{1 + \left(-\frac{\alpha - \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2}\right)}{2} \]

      9. sub-neg 91.4%

         \[\leadsto \frac{\color{blue}{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}}{2} \]

      10. div-sub 91.4%

          \[\leadsto \color{blue}{\frac{1}{2} - \frac{\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}} \]

      11. metadata-eval 91.4%

          \[\leadsto \color{blue}{0.5} - \frac{\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2} \]

      12. +-commutative 91.4%

          \[\leadsto 0.5 - \frac{\frac{\alpha - \beta}{\color{blue}{\left(\alpha + \beta\right)} + 2}}{2} \]

      13. associate-/l/ 91.4%

          \[\leadsto 0.5 - \color{blue}{\frac{\alpha - \beta}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)}} \]

      14. sub-neg 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{\alpha + \left(-\beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      15. remove-double-neg 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{\left(-\left(-\alpha\right)\right)} + \left(-\beta\right)}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      16. distribute-neg-out 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{-\left(\left(-\alpha\right) + \beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      17. neg-mul-1 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{-1 \cdot \left(\left(-\alpha\right) + \beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      18. *-commutative 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{\left(\left(-\alpha\right) + \beta\right) \cdot -1}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

   3. Simplified 91.5%

      \[\leadsto \color{blue}{0.5 - \left(\beta - \alpha\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf 89.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -4.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq -7.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 17500:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq -4.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq -2 \cdot 10^{-179}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 240000:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta -4.5e-141)
   (/ (+ 1.0 (* beta 0.5)) 2.0)
   (if (<= beta -2e-179)
     (/ (/ 2.0 alpha) 2.0)
     (if (<= beta 240000.0)
       (/ (- 1.0 (* alpha 0.5)) 2.0)
       (if (<= beta 7.2e+47) (/ (* 2.0 (/ beta alpha)) 2.0) 1.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= -4.5e-141) {
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	} else if (beta <= -2e-179) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 240000.0) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else if (beta <= 7.2e+47) {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= (-4.5d-141)) then
        tmp = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    else if (beta <= (-2d-179)) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 240000.0d0) then
        tmp = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    else if (beta <= 7.2d+47) then
        tmp = (2.0d0 * (beta / alpha)) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= -4.5e-141) {
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	} else if (beta <= -2e-179) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 240000.0) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else if (beta <= 7.2e+47) {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= -4.5e-141:
		tmp = (1.0 + (beta * 0.5)) / 2.0
	elif beta <= -2e-179:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 240000.0:
		tmp = (1.0 - (alpha * 0.5)) / 2.0
	elif beta <= 7.2e+47:
		tmp = (2.0 * (beta / alpha)) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= -4.5e-141)
		tmp = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0);
	elseif (beta <= -2e-179)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 240000.0)
		tmp = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0);
	elseif (beta <= 7.2e+47)
		tmp = Float64(Float64(2.0 * Float64(beta / alpha)) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= -4.5e-141)
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	elseif (beta <= -2e-179)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 240000.0)
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	elseif (beta <= 7.2e+47)
		tmp = (2.0 * (beta / alpha)) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, -4.5e-141], N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, -2e-179], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 240000.0], N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 7.2e+47], N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 5 regimes

2. if beta < -4.5e-141

   1. Initial program 71.1%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0 70.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]

   4. Taylor expanded in beta around 0 70.0%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \beta} + 1}{2} \]
   5. Step-by-step derivation

      1. *-commutative 70.0%

         \[\leadsto \frac{\color{blue}{\beta \cdot 0.5} + 1}{2} \]

   6. Simplified 70.0%

      \[\leadsto \frac{\color{blue}{\beta \cdot 0.5} + 1}{2} \]

   if -4.5e-141 < beta < -2e-179

   1. Initial program 31.2%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around 0 31.2%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{2 + \alpha}}}{2} \]
   4. Step-by-step derivation

      1. +-commutative 31.2%

         \[\leadsto \frac{1 - \frac{\alpha}{\color{blue}{\alpha + 2}}}{2} \]

   5. Simplified 31.2%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{\alpha + 2}}}{2} \]

   6. Taylor expanded in alpha around inf 73.4%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha}}}{2} \]

   if -2e-179 < beta < 2.4e5

   1. Initial program 75.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around 0 74.8%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{2 + \alpha}}}{2} \]
   4. Step-by-step derivation

      1. +-commutative 74.8%

         \[\leadsto \frac{1 - \frac{\alpha}{\color{blue}{\alpha + 2}}}{2} \]

   5. Simplified 74.8%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{\alpha + 2}}}{2} \]

   6. Taylor expanded in alpha around 0 72.6%

      \[\leadsto \frac{1 - \color{blue}{0.5 \cdot \alpha}}{2} \]
   7. Step-by-step derivation

      1. *-commutative 72.6%

         \[\leadsto \frac{1 - \color{blue}{\alpha \cdot 0.5}}{2} \]

   8. Simplified 72.6%

      \[\leadsto \frac{1 - \color{blue}{\alpha \cdot 0.5}}{2} \]

   if 2.4e5 < beta < 7.20000000000000015e47

   1. Initial program 26.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf 79.7%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]

   4. Taylor expanded in beta around inf 76.7%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{\beta}{\alpha}}}{2} \]

   if 7.20000000000000015e47 < beta

   1. Initial program 91.4%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 91.4%

         \[\leadsto \frac{\color{blue}{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}}{2} \]

      2. sub-neg 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{\beta + \left(-\alpha\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      3. +-commutative 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(-\alpha\right) + \beta}}{\left(\alpha + \beta\right) + 2}}{2} \]

      4. neg-sub0 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{-\left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      7. distribute-frac-neg 91.4%

         \[\leadsto \frac{1 + \color{blue}{\left(-\frac{\alpha - \beta}{\left(\alpha + \beta\right) + 2}\right)}}{2} \]

      8. +-commutative 91.4%

         \[\leadsto \frac{1 + \left(-\frac{\alpha - \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2}\right)}{2} \]

      9. sub-neg 91.4%

         \[\leadsto \frac{\color{blue}{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}}{2} \]

      10. div-sub 91.4%

          \[\leadsto \color{blue}{\frac{1}{2} - \frac{\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}} \]

      11. metadata-eval 91.4%

          \[\leadsto \color{blue}{0.5} - \frac{\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2} \]

      12. +-commutative 91.4%

          \[\leadsto 0.5 - \frac{\frac{\alpha - \beta}{\color{blue}{\left(\alpha + \beta\right)} + 2}}{2} \]

      13. associate-/l/ 91.4%

          \[\leadsto 0.5 - \color{blue}{\frac{\alpha - \beta}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)}} \]

      14. sub-neg 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{\alpha + \left(-\beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      15. remove-double-neg 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{\left(-\left(-\alpha\right)\right)} + \left(-\beta\right)}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      16. distribute-neg-out 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{-\left(\left(-\alpha\right) + \beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      17. neg-mul-1 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{-1 \cdot \left(\left(-\alpha\right) + \beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      18. *-commutative 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{\left(\left(-\alpha\right) + \beta\right) \cdot -1}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

   3. Simplified 91.5%

      \[\leadsto \color{blue}{0.5 - \left(\beta - \alpha\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf 89.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -4.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq -2 \cdot 10^{-179}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 240000:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999996)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
   (+ 0.5 (* (- alpha beta) (/ -0.5 (+ beta (+ alpha 2.0)))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999996d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = 0.5d0 + ((alpha - beta) * ((-0.5d0) / (beta + (alpha + 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999996)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) * Float64(-0.5 / Float64(beta + Float64(alpha + 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999996], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999995999999999996

   1. Initial program 5.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf 99.9%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]

   if -0.999995999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \frac{\color{blue}{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}}{2} \]

      2. sub-neg 99.8%

         \[\leadsto \frac{1 + \frac{\color{blue}{\beta + \left(-\alpha\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(-\alpha\right) + \beta}}{\left(\alpha + \beta\right) + 2}}{2} \]

      4. neg-sub0 99.8%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 99.8%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 99.8%

         \[\leadsto \frac{1 + \frac{\color{blue}{-\left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      7. distribute-frac-neg 99.8%

         \[\leadsto \frac{1 + \color{blue}{\left(-\frac{\alpha - \beta}{\left(\alpha + \beta\right) + 2}\right)}}{2} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{1 + \left(-\frac{\alpha - \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2}\right)}{2} \]

      9. sub-neg 99.8%

         \[\leadsto \frac{\color{blue}{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}}{2} \]

      10. div-sub 99.8%

          \[\leadsto \color{blue}{\frac{1}{2} - \frac{\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}} \]

      11. metadata-eval 99.8%

          \[\leadsto \color{blue}{0.5} - \frac{\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2} \]

      12. +-commutative 99.8%

          \[\leadsto 0.5 - \frac{\frac{\alpha - \beta}{\color{blue}{\left(\alpha + \beta\right)} + 2}}{2} \]

      13. associate-/l/ 99.8%

          \[\leadsto 0.5 - \color{blue}{\frac{\alpha - \beta}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)}} \]

      14. sub-neg 99.8%

          \[\leadsto 0.5 - \frac{\color{blue}{\alpha + \left(-\beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      15. remove-double-neg 99.8%

          \[\leadsto 0.5 - \frac{\color{blue}{\left(-\left(-\alpha\right)\right)} + \left(-\beta\right)}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      16. distribute-neg-out 99.8%

          \[\leadsto 0.5 - \frac{\color{blue}{-\left(\left(-\alpha\right) + \beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      17. neg-mul-1 99.8%

          \[\leadsto 0.5 - \frac{\color{blue}{-1 \cdot \left(\left(-\alpha\right) + \beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      18. *-commutative 99.8%

          \[\leadsto 0.5 - \frac{\color{blue}{\left(\left(-\alpha\right) + \beta\right) \cdot -1}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{0.5 - \left(\beta - \alpha\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.8e-23)
   (/ (/ 2.0 alpha) 2.0)
   (if (<= beta 7.2e+47) (/ (* 2.0 (/ beta alpha)) 2.0) 1.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.8e-23) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 7.2e+47) {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.8d-23) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 7.2d+47) then
        tmp = (2.0d0 * (beta / alpha)) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.8e-23) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 7.2e+47) {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 8.8e-23:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 7.2e+47:
		tmp = (2.0 * (beta / alpha)) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.8e-23)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 7.2e+47)
		tmp = Float64(Float64(2.0 * Float64(beta / alpha)) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.8e-23)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 7.2e+47)
		tmp = (2.0 * (beta / alpha)) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.8e-23], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 7.2e+47], N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if beta < 8.7999999999999998e-23

   1. Initial program 71.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around 0 70.7%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{2 + \alpha}}}{2} \]
   4. Step-by-step derivation

      1. +-commutative 70.7%

         \[\leadsto \frac{1 - \frac{\alpha}{\color{blue}{\alpha + 2}}}{2} \]

   5. Simplified 70.7%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{\alpha + 2}}}{2} \]

   6. Taylor expanded in alpha around inf 32.6%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha}}}{2} \]

   if 8.7999999999999998e-23 < beta < 7.20000000000000015e47

   1. Initial program 40.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf 64.3%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]

   4. Taylor expanded in beta around inf 61.9%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{\beta}{\alpha}}}{2} \]

   if 7.20000000000000015e47 < beta

   1. Initial program 91.4%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 91.4%

         \[\leadsto \frac{\color{blue}{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}}{2} \]

      2. sub-neg 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{\beta + \left(-\alpha\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      3. +-commutative 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(-\alpha\right) + \beta}}{\left(\alpha + \beta\right) + 2}}{2} \]

      4. neg-sub0 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 91.4%

         \[\leadsto \frac{1 + \frac{\color{blue}{-\left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      7. distribute-frac-neg 91.4%

         \[\leadsto \frac{1 + \color{blue}{\left(-\frac{\alpha - \beta}{\left(\alpha + \beta\right) + 2}\right)}}{2} \]

      8. +-commutative 91.4%

         \[\leadsto \frac{1 + \left(-\frac{\alpha - \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2}\right)}{2} \]

      9. sub-neg 91.4%

         \[\leadsto \frac{\color{blue}{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}}{2} \]

      10. div-sub 91.4%

          \[\leadsto \color{blue}{\frac{1}{2} - \frac{\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}} \]

      11. metadata-eval 91.4%

          \[\leadsto \color{blue}{0.5} - \frac{\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2} \]

      12. +-commutative 91.4%

          \[\leadsto 0.5 - \frac{\frac{\alpha - \beta}{\color{blue}{\left(\alpha + \beta\right)} + 2}}{2} \]

      13. associate-/l/ 91.4%

          \[\leadsto 0.5 - \color{blue}{\frac{\alpha - \beta}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)}} \]

      14. sub-neg 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{\alpha + \left(-\beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      15. remove-double-neg 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{\left(-\left(-\alpha\right)\right)} + \left(-\beta\right)}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      16. distribute-neg-out 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{-\left(\left(-\alpha\right) + \beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      17. neg-mul-1 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{-1 \cdot \left(\left(-\alpha\right) + \beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      18. *-commutative 91.4%

          \[\leadsto 0.5 - \frac{\color{blue}{\left(\left(-\alpha\right) + \beta\right) \cdot -1}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

   3. Simplified 91.5%

      \[\leadsto \color{blue}{0.5 - \left(\beta - \alpha\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf 89.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1600000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1600000000000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1600000000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1600000000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1600000000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1600000000000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1600000000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1600000000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1600000000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.6e12

   1. Initial program 100.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0 98.8%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]

   if 1.6e12 < alpha

   1. Initial program 27.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around 0 5.2%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{2 + \alpha}}}{2} \]
   4. Step-by-step derivation

      1. +-commutative 5.2%

         \[\leadsto \frac{1 - \frac{\alpha}{\color{blue}{\alpha + 2}}}{2} \]

   5. Simplified 5.2%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{\alpha + 2}}}{2} \]

   6. Taylor expanded in alpha around inf 57.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1600000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 4.1e+15)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 4.1e+15) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 4.1d+15) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 4.1e+15) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 4.1e+15:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 4.1e+15)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 4.1e+15)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 4.1e+15], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.1e15

   1. Initial program 100.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0 98.8%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]

   if 4.1e15 < alpha

   1. Initial program 27.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf 78.3%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.4e-23) (/ (/ 2.0 alpha) 2.0) 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.4e-23) {
		tmp = (2.0 / alpha) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.4d-23) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.4e-23) {
		tmp = (2.0 / alpha) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 8.4e-23:
		tmp = (2.0 / alpha) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.4e-23)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.4e-23)
		tmp = (2.0 / alpha) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.4e-23], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 8.4000000000000003e-23

   1. Initial program 71.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around 0 70.7%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{2 + \alpha}}}{2} \]
   4. Step-by-step derivation

      1. +-commutative 70.7%

         \[\leadsto \frac{1 - \frac{\alpha}{\color{blue}{\alpha + 2}}}{2} \]

   5. Simplified 70.7%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{\alpha + 2}}}{2} \]

   6. Taylor expanded in alpha around inf 32.6%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha}}}{2} \]

   if 8.4000000000000003e-23 < beta

   1. Initial program 84.1%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 84.1%

         \[\leadsto \frac{\color{blue}{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}}{2} \]

      2. sub-neg 84.1%

         \[\leadsto \frac{1 + \frac{\color{blue}{\beta + \left(-\alpha\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      3. +-commutative 84.1%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(-\alpha\right) + \beta}}{\left(\alpha + \beta\right) + 2}}{2} \]

      4. neg-sub0 84.1%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 84.1%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 84.1%

         \[\leadsto \frac{1 + \frac{\color{blue}{-\left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      7. distribute-frac-neg 84.1%

         \[\leadsto \frac{1 + \color{blue}{\left(-\frac{\alpha - \beta}{\left(\alpha + \beta\right) + 2}\right)}}{2} \]

      8. +-commutative 84.1%

         \[\leadsto \frac{1 + \left(-\frac{\alpha - \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2}\right)}{2} \]

      9. sub-neg 84.1%

         \[\leadsto \frac{\color{blue}{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}}{2} \]

      10. div-sub 84.1%

          \[\leadsto \color{blue}{\frac{1}{2} - \frac{\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}} \]

      11. metadata-eval 84.1%

          \[\leadsto \color{blue}{0.5} - \frac{\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2} \]

      12. +-commutative 84.1%

          \[\leadsto 0.5 - \frac{\frac{\alpha - \beta}{\color{blue}{\left(\alpha + \beta\right)} + 2}}{2} \]

      13. associate-/l/ 84.1%

          \[\leadsto 0.5 - \color{blue}{\frac{\alpha - \beta}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)}} \]

      14. sub-neg 84.1%

          \[\leadsto 0.5 - \frac{\color{blue}{\alpha + \left(-\beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      15. remove-double-neg 84.1%

          \[\leadsto 0.5 - \frac{\color{blue}{\left(-\left(-\alpha\right)\right)} + \left(-\beta\right)}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      16. distribute-neg-out 84.1%

          \[\leadsto 0.5 - \frac{\color{blue}{-\left(\left(-\alpha\right) + \beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      17. neg-mul-1 84.1%

          \[\leadsto 0.5 - \frac{\color{blue}{-1 \cdot \left(\left(-\alpha\right) + \beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

      18. *-commutative 84.1%

          \[\leadsto 0.5 - \frac{\color{blue}{\left(\left(-\alpha\right) + \beta\right) \cdot -1}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

   3. Simplified 84.2%

      \[\leadsto \color{blue}{0.5 - \left(\beta - \alpha\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf 80.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.1% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 76.3%

   \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
2. Step-by-step derivation

   1. +-commutative 76.3%

      \[\leadsto \frac{\color{blue}{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}}{2} \]

   2. sub-neg 76.3%

      \[\leadsto \frac{1 + \frac{\color{blue}{\beta + \left(-\alpha\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

   3. +-commutative 76.3%

      \[\leadsto \frac{1 + \frac{\color{blue}{\left(-\alpha\right) + \beta}}{\left(\alpha + \beta\right) + 2}}{2} \]

   4. neg-sub0 76.3%

      \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

   5. associate-+l- 76.3%

      \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

   6. sub0-neg 76.3%

      \[\leadsto \frac{1 + \frac{\color{blue}{-\left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

   7. distribute-frac-neg 76.3%

      \[\leadsto \frac{1 + \color{blue}{\left(-\frac{\alpha - \beta}{\left(\alpha + \beta\right) + 2}\right)}}{2} \]

   8. +-commutative 76.3%

      \[\leadsto \frac{1 + \left(-\frac{\alpha - \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2}\right)}{2} \]

   9. sub-neg 76.3%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}}{2} \]

   10. div-sub 76.3%

       \[\leadsto \color{blue}{\frac{1}{2} - \frac{\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}} \]

   11. metadata-eval 76.3%

       \[\leadsto \color{blue}{0.5} - \frac{\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2} \]

   12. +-commutative 76.3%

       \[\leadsto 0.5 - \frac{\frac{\alpha - \beta}{\color{blue}{\left(\alpha + \beta\right)} + 2}}{2} \]

   13. associate-/l/ 76.3%

       \[\leadsto 0.5 - \color{blue}{\frac{\alpha - \beta}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)}} \]

   14. sub-neg 76.3%

       \[\leadsto 0.5 - \frac{\color{blue}{\alpha + \left(-\beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

   15. remove-double-neg 76.3%

       \[\leadsto 0.5 - \frac{\color{blue}{\left(-\left(-\alpha\right)\right)} + \left(-\beta\right)}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

   16. distribute-neg-out 76.3%

       \[\leadsto 0.5 - \frac{\color{blue}{-\left(\left(-\alpha\right) + \beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

   17. neg-mul-1 76.3%

       \[\leadsto 0.5 - \frac{\color{blue}{-1 \cdot \left(\left(-\alpha\right) + \beta\right)}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

   18. *-commutative 76.3%

       \[\leadsto 0.5 - \frac{\color{blue}{\left(\left(-\alpha\right) + \beta\right) \cdot -1}}{2 \cdot \left(\left(\alpha + \beta\right) + 2\right)} \]

3. Simplified 76.3%

   \[\leadsto \color{blue}{0.5 - \left(\beta - \alpha\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in beta around inf 41.1%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 41.1%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(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))