Octave 3.8, jcobi/1

Percentage Accurate: 74.6% → 99.7%

Time: 10.1s

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: 74.6% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999999)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
   (/ (exp (log1p (/ (- beta alpha) (+ beta (+ alpha 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = exp(log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = Math.exp(Math.log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = math.exp(math.log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999999)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(exp(log1p(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999999], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[Log[1 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999998999999999971

   1. Initial program 5.9%

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

      1. +-commutative 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in alpha around -inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. sub-neg 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. neg-mul-1 99.4%

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

      5. mul-1-neg 99.4%

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

      6. remove-double-neg 99.4%

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

      7. neg-mul-1 99.4%

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

      8. remove-double-neg 99.4%

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

   7. Simplified 99.4%

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

   if -0.999998999999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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%

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

   3. Simplified 99.8%

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

      1. add-exp-log 99.8%

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

      2. +-commutative 99.8%

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

      3. log1p-define 99.8%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}}}{2} \]

      4. associate-+l+ 99.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right)}}{2} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999999)
     (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 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) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.999999d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.999999:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.999999)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.999999)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

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, -0.999999], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999998999999999971

   1. Initial program 5.9%

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

      1. +-commutative 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in alpha around -inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. sub-neg 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. neg-mul-1 99.4%

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

      5. mul-1-neg 99.4%

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

      6. remove-double-neg 99.4%

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

      7. neg-mul-1 99.4%

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

      8. remove-double-neg 99.4%

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

   7. Simplified 99.4%

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

   if -0.999998999999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \beta \cdot \left(0.25 + \beta \cdot -0.125\right)\\ \mathbf{if}\;\beta \leq -1.2 \cdot 10^{-100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq -6.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{1 + \frac{-2}{\alpha}}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* beta (+ 0.25 (* beta -0.125))))))
   (if (<= beta -1.2e-100)
     t_0
     (if (<= beta -6.5e-178)
       (/ (+ 1.0 (/ -2.0 alpha)) alpha)
       (if (<= beta 2.0) t_0 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (beta * (0.25 + (beta * -0.125)));
	double tmp;
	if (beta <= -1.2e-100) {
		tmp = t_0;
	} else if (beta <= -6.5e-178) {
		tmp = (1.0 + (-2.0 / alpha)) / alpha;
	} else if (beta <= 2.0) {
		tmp = t_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 = 0.5d0 + (beta * (0.25d0 + (beta * (-0.125d0))))
    if (beta <= (-1.2d-100)) then
        tmp = t_0
    else if (beta <= (-6.5d-178)) then
        tmp = (1.0d0 + ((-2.0d0) / alpha)) / alpha
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (beta * (0.25 + (beta * -0.125)));
	double tmp;
	if (beta <= -1.2e-100) {
		tmp = t_0;
	} else if (beta <= -6.5e-178) {
		tmp = (1.0 + (-2.0 / alpha)) / alpha;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (beta * (0.25 + (beta * -0.125)))
	tmp = 0
	if beta <= -1.2e-100:
		tmp = t_0
	elif beta <= -6.5e-178:
		tmp = (1.0 + (-2.0 / alpha)) / alpha
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(beta * Float64(0.25 + Float64(beta * -0.125))))
	tmp = 0.0
	if (beta <= -1.2e-100)
		tmp = t_0;
	elseif (beta <= -6.5e-178)
		tmp = Float64(Float64(1.0 + Float64(-2.0 / alpha)) / alpha);
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (beta * (0.25 + (beta * -0.125)));
	tmp = 0.0;
	if (beta <= -1.2e-100)
		tmp = t_0;
	elseif (beta <= -6.5e-178)
		tmp = (1.0 + (-2.0 / alpha)) / alpha;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(beta * N[(0.25 + N[(beta * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, -1.2e-100], t$95$0, If[LessEqual[beta, -6.5e-178], N[(N[(1.0 + N[(-2.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[beta, 2.0], t$95$0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if beta < -1.2000000000000001e-100 or -6.5000000000000002e-178 < beta < 2

   1. Initial program 77.4%

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

      1. +-commutative 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in alpha around 0 73.7%

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

   6. Taylor expanded in beta around 0 73.4%

      \[\leadsto \frac{\color{blue}{\beta \cdot \left(0.5 + -0.25 \cdot \beta\right)} + 1}{2} \]
   7. Step-by-step derivation

      1. *-commutative 73.4%

         \[\leadsto \frac{\beta \cdot \left(0.5 + \color{blue}{\beta \cdot -0.25}\right) + 1}{2} \]

   8. Simplified 73.4%

      \[\leadsto \frac{\color{blue}{\beta \cdot \left(0.5 + \beta \cdot -0.25\right)} + 1}{2} \]

   9. Taylor expanded in beta around 0 73.4%

      \[\leadsto \color{blue}{0.5 + \beta \cdot \left(0.25 + -0.125 \cdot \beta\right)} \]
   10. Step-by-step derivation

       1. *-commutative 73.4%

          \[\leadsto 0.5 + \beta \cdot \left(0.25 + \color{blue}{\beta \cdot -0.125}\right) \]

   11. Simplified 73.4%

       \[\leadsto \color{blue}{0.5 + \beta \cdot \left(0.25 + \beta \cdot -0.125\right)} \]

   if -1.2000000000000001e-100 < beta < -6.5000000000000002e-178

   1. Initial program 39.5%

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

      1. +-commutative 39.5%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in beta around 0 39.5%

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

      1. +-commutative 39.5%

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

   7. Simplified 39.5%

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

   8. Taylor expanded in alpha around inf 67.0%

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

      1. associate-*r/ 67.0%

         \[\leadsto \frac{\frac{2 - \color{blue}{\frac{4 \cdot 1}{\alpha}}}{\alpha}}{2} \]

      2. metadata-eval 67.0%

         \[\leadsto \frac{\frac{2 - \frac{\color{blue}{4}}{\alpha}}{\alpha}}{2} \]

   10. Simplified 67.0%

       \[\leadsto \frac{\color{blue}{\frac{2 - \frac{4}{\alpha}}{\alpha}}}{2} \]

   11. Taylor expanded in alpha around inf 67.0%

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

       1. sub-neg 67.0%

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

       2. associate-*r/ 67.0%

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

       3. metadata-eval 67.0%

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

       4. distribute-neg-frac 67.0%

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

       5. metadata-eval 67.0%

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

   13. Simplified 67.0%

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

   if 2 < beta

   1. Initial program 79.7%

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

      1. +-commutative 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in beta around inf 78.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -1.2 \cdot 10^{-100}:\\ \;\;\;\;0.5 + \beta \cdot \left(0.25 + \beta \cdot -0.125\right)\\ \mathbf{elif}\;\beta \leq -6.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{1 + \frac{-2}{\alpha}}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot \left(0.25 + \beta \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 37:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.9 \cdot 10^{+210} \lor \neg \left(\alpha \leq 2.8 \cdot 10^{+262}\right):\\ \;\;\;\;\frac{1 + \frac{-2}{\alpha}}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 37.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (or (<= alpha 3.9e+210) (not (<= alpha 2.8e+262)))
     (/ (+ 1.0 (/ -2.0 alpha)) alpha)
     (/ (/ (* beta 2.0) alpha) 2.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 37.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 3.9e+210) || !(alpha <= 2.8e+262)) {
		tmp = (1.0 + (-2.0 / alpha)) / alpha;
	} else {
		tmp = ((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 <= 37.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if ((alpha <= 3.9d+210) .or. (.not. (alpha <= 2.8d+262))) then
        tmp = (1.0d0 + ((-2.0d0) / alpha)) / alpha
    else
        tmp = ((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 <= 37.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 3.9e+210) || !(alpha <= 2.8e+262)) {
		tmp = (1.0 + (-2.0 / alpha)) / alpha;
	} else {
		tmp = ((beta * 2.0) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 37.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif (alpha <= 3.9e+210) or not (alpha <= 2.8e+262):
		tmp = (1.0 + (-2.0 / alpha)) / alpha
	else:
		tmp = ((beta * 2.0) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 37.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif ((alpha <= 3.9e+210) || !(alpha <= 2.8e+262))
		tmp = Float64(Float64(1.0 + Float64(-2.0 / alpha)) / alpha);
	else
		tmp = Float64(Float64(Float64(beta * 2.0) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 37.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif ((alpha <= 3.9e+210) || ~((alpha <= 2.8e+262)))
		tmp = (1.0 + (-2.0 / alpha)) / alpha;
	else
		tmp = ((beta * 2.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 37.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 3.9e+210], N[Not[LessEqual[alpha, 2.8e+262]], $MachinePrecision]], N[(N[(1.0 + N[(-2.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta * 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 37

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.3%

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

   if 37 < alpha < 3.9e210 or 2.79999999999999998e262 < alpha

   1. Initial program 28.0%

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

      1. +-commutative 28.0%

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

   3. Simplified 28.0%

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

   5. Taylor expanded in beta around 0 9.0%

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

      1. +-commutative 9.0%

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

   7. Simplified 9.0%

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

   8. Taylor expanded in alpha around inf 63.8%

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

      1. associate-*r/ 63.8%

         \[\leadsto \frac{\frac{2 - \color{blue}{\frac{4 \cdot 1}{\alpha}}}{\alpha}}{2} \]

      2. metadata-eval 63.8%

         \[\leadsto \frac{\frac{2 - \frac{\color{blue}{4}}{\alpha}}{\alpha}}{2} \]

   10. Simplified 63.8%

       \[\leadsto \frac{\color{blue}{\frac{2 - \frac{4}{\alpha}}{\alpha}}}{2} \]

   11. Taylor expanded in alpha around inf 63.8%

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

       1. sub-neg 63.8%

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

       2. associate-*r/ 63.8%

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

       3. metadata-eval 63.8%

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

       4. distribute-neg-frac 63.8%

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

       5. metadata-eval 63.8%

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

   13. Simplified 63.8%

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

   if 3.9e210 < alpha < 2.79999999999999998e262

   1. Initial program 17.7%

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

      1. +-commutative 17.7%

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

   3. Simplified 17.7%

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

   5. Taylor expanded in alpha around -inf 87.8%

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

      1. associate-*r/ 87.8%

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

      2. sub-neg 87.8%

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

      3. distribute-lft-in 87.8%

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

      4. neg-mul-1 87.8%

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

      5. mul-1-neg 87.8%

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

      6. remove-double-neg 87.8%

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

      7. neg-mul-1 87.8%

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

      8. remove-double-neg 87.8%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in beta around inf 62.8%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 37:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.9 \cdot 10^{+210} \lor \neg \left(\alpha \leq 2.8 \cdot 10^{+262}\right):\\ \;\;\;\;\frac{1 + \frac{-2}{\alpha}}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \beta \cdot 0.25\\ \mathbf{if}\;\beta \leq -1.22 \cdot 10^{-100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq -9.6 \cdot 10^{-178}:\\ \;\;\;\;\frac{1 + \frac{-2}{\alpha}}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* beta 0.25))))
   (if (<= beta -1.22e-100)
     t_0
     (if (<= beta -9.6e-178)
       (/ (+ 1.0 (/ -2.0 alpha)) alpha)
       (if (<= beta 2.0) t_0 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (beta * 0.25);
	double tmp;
	if (beta <= -1.22e-100) {
		tmp = t_0;
	} else if (beta <= -9.6e-178) {
		tmp = (1.0 + (-2.0 / alpha)) / alpha;
	} else if (beta <= 2.0) {
		tmp = t_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 = 0.5d0 + (beta * 0.25d0)
    if (beta <= (-1.22d-100)) then
        tmp = t_0
    else if (beta <= (-9.6d-178)) then
        tmp = (1.0d0 + ((-2.0d0) / alpha)) / alpha
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (beta * 0.25);
	double tmp;
	if (beta <= -1.22e-100) {
		tmp = t_0;
	} else if (beta <= -9.6e-178) {
		tmp = (1.0 + (-2.0 / alpha)) / alpha;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (beta * 0.25)
	tmp = 0
	if beta <= -1.22e-100:
		tmp = t_0
	elif beta <= -9.6e-178:
		tmp = (1.0 + (-2.0 / alpha)) / alpha
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(beta * 0.25))
	tmp = 0.0
	if (beta <= -1.22e-100)
		tmp = t_0;
	elseif (beta <= -9.6e-178)
		tmp = Float64(Float64(1.0 + Float64(-2.0 / alpha)) / alpha);
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (beta * 0.25);
	tmp = 0.0;
	if (beta <= -1.22e-100)
		tmp = t_0;
	elseif (beta <= -9.6e-178)
		tmp = (1.0 + (-2.0 / alpha)) / alpha;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, -1.22e-100], t$95$0, If[LessEqual[beta, -9.6e-178], N[(N[(1.0 + N[(-2.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[beta, 2.0], t$95$0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if beta < -1.2199999999999999e-100 or -9.6000000000000002e-178 < beta < 2

   1. Initial program 77.4%

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

      1. +-commutative 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in alpha around 0 73.7%

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

   6. Taylor expanded in beta around 0 73.4%

      \[\leadsto \frac{\color{blue}{\beta \cdot \left(0.5 + -0.25 \cdot \beta\right)} + 1}{2} \]
   7. Step-by-step derivation

      1. *-commutative 73.4%

         \[\leadsto \frac{\beta \cdot \left(0.5 + \color{blue}{\beta \cdot -0.25}\right) + 1}{2} \]

   8. Simplified 73.4%

      \[\leadsto \frac{\color{blue}{\beta \cdot \left(0.5 + \beta \cdot -0.25\right)} + 1}{2} \]

   9. Taylor expanded in beta around 0 73.0%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \beta} \]

   if -1.2199999999999999e-100 < beta < -9.6000000000000002e-178

   1. Initial program 39.5%

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

      1. +-commutative 39.5%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in beta around 0 39.5%

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

      1. +-commutative 39.5%

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

   7. Simplified 39.5%

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

   8. Taylor expanded in alpha around inf 67.0%

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

      1. associate-*r/ 67.0%

         \[\leadsto \frac{\frac{2 - \color{blue}{\frac{4 \cdot 1}{\alpha}}}{\alpha}}{2} \]

      2. metadata-eval 67.0%

         \[\leadsto \frac{\frac{2 - \frac{\color{blue}{4}}{\alpha}}{\alpha}}{2} \]

   10. Simplified 67.0%

       \[\leadsto \frac{\color{blue}{\frac{2 - \frac{4}{\alpha}}{\alpha}}}{2} \]

   11. Taylor expanded in alpha around inf 67.0%

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

       1. sub-neg 67.0%

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

       2. associate-*r/ 67.0%

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

       3. metadata-eval 67.0%

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

       4. distribute-neg-frac 67.0%

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

       5. metadata-eval 67.0%

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

   13. Simplified 67.0%

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

   if 2 < beta

   1. Initial program 79.7%

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

      1. +-commutative 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in beta around inf 78.1%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -1.22 \cdot 10^{-100}:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{elif}\;\beta \leq -9.6 \cdot 10^{-178}:\\ \;\;\;\;\frac{1 + \frac{-2}{\alpha}}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \beta \cdot 0.25\\ \mathbf{if}\;\beta \leq -1.25 \cdot 10^{-100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq -9.6 \cdot 10^{-178}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* beta 0.25))))
   (if (<= beta -1.25e-100)
     t_0
     (if (<= beta -9.6e-178) (/ 1.0 alpha) (if (<= beta 2.0) t_0 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (beta * 0.25);
	double tmp;
	if (beta <= -1.25e-100) {
		tmp = t_0;
	} else if (beta <= -9.6e-178) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = t_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 = 0.5d0 + (beta * 0.25d0)
    if (beta <= (-1.25d-100)) then
        tmp = t_0
    else if (beta <= (-9.6d-178)) then
        tmp = 1.0d0 / alpha
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (beta * 0.25);
	double tmp;
	if (beta <= -1.25e-100) {
		tmp = t_0;
	} else if (beta <= -9.6e-178) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (beta * 0.25)
	tmp = 0
	if beta <= -1.25e-100:
		tmp = t_0
	elif beta <= -9.6e-178:
		tmp = 1.0 / alpha
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(beta * 0.25))
	tmp = 0.0
	if (beta <= -1.25e-100)
		tmp = t_0;
	elseif (beta <= -9.6e-178)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (beta * 0.25);
	tmp = 0.0;
	if (beta <= -1.25e-100)
		tmp = t_0;
	elseif (beta <= -9.6e-178)
		tmp = 1.0 / alpha;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, -1.25e-100], t$95$0, If[LessEqual[beta, -9.6e-178], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 2.0], t$95$0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if beta < -1.25e-100 or -9.6000000000000002e-178 < beta < 2

   1. Initial program 77.4%

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

      1. +-commutative 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in alpha around 0 73.7%

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

   6. Taylor expanded in beta around 0 73.4%

      \[\leadsto \frac{\color{blue}{\beta \cdot \left(0.5 + -0.25 \cdot \beta\right)} + 1}{2} \]
   7. Step-by-step derivation

      1. *-commutative 73.4%

         \[\leadsto \frac{\beta \cdot \left(0.5 + \color{blue}{\beta \cdot -0.25}\right) + 1}{2} \]

   8. Simplified 73.4%

      \[\leadsto \frac{\color{blue}{\beta \cdot \left(0.5 + \beta \cdot -0.25\right)} + 1}{2} \]

   9. Taylor expanded in beta around 0 73.0%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \beta} \]

   if -1.25e-100 < beta < -9.6000000000000002e-178

   1. Initial program 39.5%

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

      1. +-commutative 39.5%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in beta around 0 39.5%

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

      1. +-commutative 39.5%

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

   7. Simplified 39.5%

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

   8. Taylor expanded in alpha around inf 67.0%

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

      1. associate-*r/ 67.0%

         \[\leadsto \frac{\frac{2 - \color{blue}{\frac{4 \cdot 1}{\alpha}}}{\alpha}}{2} \]

      2. metadata-eval 67.0%

         \[\leadsto \frac{\frac{2 - \frac{\color{blue}{4}}{\alpha}}{\alpha}}{2} \]

   10. Simplified 67.0%

       \[\leadsto \frac{\color{blue}{\frac{2 - \frac{4}{\alpha}}{\alpha}}}{2} \]

   11. Taylor expanded in alpha around inf 65.9%

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

   if 2 < beta

   1. Initial program 79.7%

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

      1. +-commutative 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in beta around inf 78.1%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -1.25 \cdot 10^{-100}:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{elif}\;\beta \leq -9.6 \cdot 10^{-178}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq -1.2 \cdot 10^{-100}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -6.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 1750:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta -1.2e-100)
   0.5
   (if (<= beta -6.5e-178) (/ 1.0 alpha) (if (<= beta 1750.0) 0.5 1.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= -1.2e-100) {
		tmp = 0.5;
	} else if (beta <= -6.5e-178) {
		tmp = 1.0 / alpha;
	} else if (beta <= 1750.0) {
		tmp = 0.5;
	} 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 <= (-1.2d-100)) then
        tmp = 0.5d0
    else if (beta <= (-6.5d-178)) then
        tmp = 1.0d0 / alpha
    else if (beta <= 1750.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= -1.2e-100) {
		tmp = 0.5;
	} else if (beta <= -6.5e-178) {
		tmp = 1.0 / alpha;
	} else if (beta <= 1750.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= -1.2e-100:
		tmp = 0.5
	elif beta <= -6.5e-178:
		tmp = 1.0 / alpha
	elif beta <= 1750.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= -1.2e-100)
		tmp = 0.5;
	elseif (beta <= -6.5e-178)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 1750.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= -1.2e-100)
		tmp = 0.5;
	elseif (beta <= -6.5e-178)
		tmp = 1.0 / alpha;
	elseif (beta <= 1750.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, -1.2e-100], 0.5, If[LessEqual[beta, -6.5e-178], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 1750.0], 0.5, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if beta < -1.2000000000000001e-100 or -6.5000000000000002e-178 < beta < 1750

   1. Initial program 76.9%

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

      1. +-commutative 76.9%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in alpha around 0 73.2%

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

   6. Taylor expanded in beta around 0 72.9%

      \[\leadsto \frac{\color{blue}{\beta \cdot \left(0.5 + -0.25 \cdot \beta\right)} + 1}{2} \]
   7. Step-by-step derivation

      1. *-commutative 72.9%

         \[\leadsto \frac{\beta \cdot \left(0.5 + \color{blue}{\beta \cdot -0.25}\right) + 1}{2} \]

   8. Simplified 72.9%

      \[\leadsto \frac{\color{blue}{\beta \cdot \left(0.5 + \beta \cdot -0.25\right)} + 1}{2} \]

   9. Taylor expanded in beta around 0 71.5%

      \[\leadsto \color{blue}{0.5} \]

   if -1.2000000000000001e-100 < beta < -6.5000000000000002e-178

   1. Initial program 39.5%

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

      1. +-commutative 39.5%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in beta around 0 39.5%

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

      1. +-commutative 39.5%

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

   7. Simplified 39.5%

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

   8. Taylor expanded in alpha around inf 67.0%

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

      1. associate-*r/ 67.0%

         \[\leadsto \frac{\frac{2 - \color{blue}{\frac{4 \cdot 1}{\alpha}}}{\alpha}}{2} \]

      2. metadata-eval 67.0%

         \[\leadsto \frac{\frac{2 - \frac{\color{blue}{4}}{\alpha}}{\alpha}}{2} \]

   10. Simplified 67.0%

       \[\leadsto \frac{\color{blue}{\frac{2 - \frac{4}{\alpha}}{\alpha}}}{2} \]

   11. Taylor expanded in alpha around inf 65.9%

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

   if 1750 < beta

   1. Initial program 80.4%

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

      1. +-commutative 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in beta around inf 78.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -1.2 \cdot 10^{-100}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -6.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 1750:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{+52}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.9e+52)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.9e+52) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (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 <= 2.9d+52) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + (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 <= 2.9e+52) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.9e+52:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.9e+52)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.9e+52)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.9e+52], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.9e52

   1. Initial program 98.5%

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

      1. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in alpha around 0 95.5%

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

   if 2.9e52 < alpha

   1. Initial program 20.1%

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

      1. +-commutative 20.1%

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

   3. Simplified 20.1%

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

   5. Taylor expanded in alpha around -inf 85.6%

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

      1. associate-*r/ 85.6%

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

      2. sub-neg 85.6%

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

      3. distribute-lft-in 85.6%

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

      4. neg-mul-1 85.6%

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

      5. mul-1-neg 85.6%

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

      6. remove-double-neg 85.6%

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

      7. neg-mul-1 85.6%

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

      8. remove-double-neg 85.6%

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

   7. Simplified 85.6%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{+52}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.95:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.95) 0.5 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.95) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.95d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.95) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.95:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.95)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.95)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.95], 0.5, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.94999999999999996

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.3%

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

   6. Taylor expanded in beta around 0 61.9%

      \[\leadsto \frac{\color{blue}{\beta \cdot \left(0.5 + -0.25 \cdot \beta\right)} + 1}{2} \]
   7. Step-by-step derivation

      1. *-commutative 61.9%

         \[\leadsto \frac{\beta \cdot \left(0.5 + \color{blue}{\beta \cdot -0.25}\right) + 1}{2} \]

   8. Simplified 61.9%

      \[\leadsto \frac{\color{blue}{\beta \cdot \left(0.5 + \beta \cdot -0.25\right)} + 1}{2} \]

   9. Taylor expanded in beta around 0 67.4%

      \[\leadsto \color{blue}{0.5} \]

   if 1.94999999999999996 < alpha

   1. Initial program 26.2%

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

      1. +-commutative 26.2%

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

   3. Simplified 26.2%

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

   5. Taylor expanded in beta around 0 8.3%

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

      1. +-commutative 8.3%

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

   7. Simplified 8.3%

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

   8. Taylor expanded in alpha around inf 58.0%

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

      1. associate-*r/ 58.0%

         \[\leadsto \frac{\frac{2 - \color{blue}{\frac{4 \cdot 1}{\alpha}}}{\alpha}}{2} \]

      2. metadata-eval 58.0%

         \[\leadsto \frac{\frac{2 - \frac{\color{blue}{4}}{\alpha}}{\alpha}}{2} \]

   10. Simplified 58.0%

       \[\leadsto \frac{\color{blue}{\frac{2 - \frac{4}{\alpha}}{\alpha}}}{2} \]

   11. Taylor expanded in alpha around inf 56.9%

       \[\leadsto \color{blue}{\frac{1}{\alpha}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 49.2% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

double code(double alpha, double beta) {
	return 0.5;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.5d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.5;
}

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.5;
end

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 75.2%

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

   1. +-commutative 75.2%

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

3. Simplified 75.2%

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

5. Taylor expanded in alpha around 0 72.3%

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

6. Taylor expanded in beta around 0 42.4%

   \[\leadsto \frac{\color{blue}{\beta \cdot \left(0.5 + -0.25 \cdot \beta\right)} + 1}{2} \]
7. Step-by-step derivation

   1. *-commutative 42.4%

      \[\leadsto \frac{\beta \cdot \left(0.5 + \color{blue}{\beta \cdot -0.25}\right) + 1}{2} \]

8. Simplified 42.4%

   \[\leadsto \frac{\color{blue}{\beta \cdot \left(0.5 + \beta \cdot -0.25\right)} + 1}{2} \]

9. Taylor expanded in beta around 0 47.5%

   \[\leadsto \color{blue}{0.5} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024094 
(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))