Octave 3.8, jcobi/1

Percentage Accurate: 74.5% → 99.9%

Time: 17.7s

Alternatives: 12

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: 74.5% 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha} - \frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{\frac{\alpha}{\beta + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9998)
   (/
    (-
     (/ (+ 2.0 (* beta 2.0)) alpha)
     (/ (/ (+ beta (+ beta 2.0)) alpha) (/ alpha (+ beta 2.0))))
    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)) <= -0.9998) {
		tmp = (((2.0 + (beta * 2.0)) / alpha) - (((beta + (beta + 2.0)) / alpha) / (alpha / (beta + 2.0)))) / 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)) <= -0.9998) {
		tmp = (((2.0 + (beta * 2.0)) / alpha) - (((beta + (beta + 2.0)) / alpha) / (alpha / (beta + 2.0)))) / 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)) <= -0.9998:
		tmp = (((2.0 + (beta * 2.0)) / alpha) - (((beta + (beta + 2.0)) / alpha) / (alpha / (beta + 2.0)))) / 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)) <= -0.9998)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) - Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / Float64(alpha / Float64(beta + 2.0)))) / 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], -0.9998], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / N[(alpha / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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) 2)) < -0.99980000000000002

   1. Initial program 7.6%

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

      1. flip3-+ 7.7%

         \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)}}}{2} \]

      2. add-exp-log 7.7%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)}\right)}}}{2} \]

      3. flip3-+ 7.6%

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

      4. +-commutative 7.6%

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

      5. log1p-udef 7.6%

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

      6. +-commutative 7.6%

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

      7. associate-+l+ 7.6%

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

   4. Applied egg-rr 7.6%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}}{2} \]
   5. Step-by-step derivation

      1. +-commutative 7.6%

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

      2. +-commutative 7.6%

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

      3. +-commutative 7.6%

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

   6. Simplified 7.6%

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

   7. Taylor expanded in alpha around inf 90.2%

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

      1. mul-1-neg 90.2%

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

      2. unsub-neg 90.2%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in beta around 0 99.9%

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

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

      1. flip3-+ 99.9%

         \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)}}}{2} \]

      2. add-exp-log 99.9%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)}\right)}}}{2} \]

      3. flip3-+ 99.9%

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

      4. +-commutative 99.9%

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

      5. log1p-udef 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}}{2} \]
   5. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha} - \frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{\frac{\alpha}{\beta + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha} - \frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{\frac{\alpha}{\beta + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9998)
     (/
      (-
       (/ (+ 2.0 (* beta 2.0)) alpha)
       (/ (/ (+ beta (+ beta 2.0)) alpha) (/ alpha (+ beta 2.0))))
      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 <= -0.9998) {
		tmp = (((2.0 + (beta * 2.0)) / alpha) - (((beta + (beta + 2.0)) / alpha) / (alpha / (beta + 2.0)))) / 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 <= (-0.9998d0)) then
        tmp = (((2.0d0 + (beta * 2.0d0)) / alpha) - (((beta + (beta + 2.0d0)) / alpha) / (alpha / (beta + 2.0d0)))) / 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 <= -0.9998) {
		tmp = (((2.0 + (beta * 2.0)) / alpha) - (((beta + (beta + 2.0)) / alpha) / (alpha / (beta + 2.0)))) / 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 <= -0.9998:
		tmp = (((2.0 + (beta * 2.0)) / alpha) - (((beta + (beta + 2.0)) / alpha) / (alpha / (beta + 2.0)))) / 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 <= -0.9998)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) - Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / Float64(alpha / Float64(beta + 2.0)))) / 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 <= -0.9998)
		tmp = (((2.0 + (beta * 2.0)) / alpha) - (((beta + (beta + 2.0)) / alpha) / (alpha / (beta + 2.0)))) / 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, -0.9998], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / N[(alpha / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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) 2)) < -0.99980000000000002

   1. Initial program 7.6%

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

      1. flip3-+ 7.7%

         \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)}}}{2} \]

      2. add-exp-log 7.7%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)}\right)}}}{2} \]

      3. flip3-+ 7.6%

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

      4. +-commutative 7.6%

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

      5. log1p-udef 7.6%

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

      6. +-commutative 7.6%

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

      7. associate-+l+ 7.6%

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

   4. Applied egg-rr 7.6%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}}{2} \]
   5. Step-by-step derivation

      1. +-commutative 7.6%

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

      2. +-commutative 7.6%

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

      3. +-commutative 7.6%

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

   6. Simplified 7.6%

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

   7. Taylor expanded in alpha around inf 90.2%

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

      1. mul-1-neg 90.2%

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

      2. unsub-neg 90.2%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in beta around 0 99.9%

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

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha} - \frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{\frac{\alpha}{\beta + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ t_1 := \frac{\beta + \left(\beta + 2\right)}{\alpha}\\ \mathbf{if}\;t\_0 \leq -0.9998:\\ \;\;\;\;\frac{t\_1 - \frac{t\_1}{\alpha \cdot 0.5}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0)))
        (t_1 (/ (+ beta (+ beta 2.0)) alpha)))
   (if (<= t_0 -0.9998)
     (/ (- t_1 (/ t_1 (* alpha 0.5))) 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 t_1 = (beta + (beta + 2.0)) / alpha;
	double tmp;
	if (t_0 <= -0.9998) {
		tmp = (t_1 - (t_1 / (alpha * 0.5))) / 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) :: t_1
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    t_1 = (beta + (beta + 2.0d0)) / alpha
    if (t_0 <= (-0.9998d0)) then
        tmp = (t_1 - (t_1 / (alpha * 0.5d0))) / 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 t_1 = (beta + (beta + 2.0)) / alpha;
	double tmp;
	if (t_0 <= -0.9998) {
		tmp = (t_1 - (t_1 / (alpha * 0.5))) / 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)
	t_1 = (beta + (beta + 2.0)) / alpha
	tmp = 0
	if t_0 <= -0.9998:
		tmp = (t_1 - (t_1 / (alpha * 0.5))) / 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))
	t_1 = Float64(Float64(beta + Float64(beta + 2.0)) / alpha)
	tmp = 0.0
	if (t_0 <= -0.9998)
		tmp = Float64(Float64(t_1 - Float64(t_1 / Float64(alpha * 0.5))) / 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);
	t_1 = (beta + (beta + 2.0)) / alpha;
	tmp = 0.0;
	if (t_0 <= -0.9998)
		tmp = (t_1 - (t_1 / (alpha * 0.5))) / 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]}, Block[{t$95$1 = N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9998], N[(N[(t$95$1 - N[(t$95$1 / N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision]), $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) 2)) < -0.99980000000000002

   1. Initial program 7.6%

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

      1. flip3-+ 7.7%

         \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)}}}{2} \]

      2. add-exp-log 7.7%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)}\right)}}}{2} \]

      3. flip3-+ 7.6%

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

      4. +-commutative 7.6%

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

      5. log1p-udef 7.6%

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

      6. +-commutative 7.6%

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

      7. associate-+l+ 7.6%

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

   4. Applied egg-rr 7.6%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}}{2} \]
   5. Step-by-step derivation

      1. +-commutative 7.6%

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

      2. +-commutative 7.6%

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

      3. +-commutative 7.6%

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

   6. Simplified 7.6%

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

   7. Taylor expanded in alpha around inf 90.2%

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

      1. mul-1-neg 90.2%

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

      2. unsub-neg 90.2%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in beta around 0 99.2%

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

       1. *-commutative 99.2%

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

   12. Simplified 99.2%

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

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha} - \frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{\alpha \cdot 0.5}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ t_1 := \frac{\beta + \left(\beta + 2\right)}{\alpha}\\ \mathbf{if}\;t\_0 \leq -0.9998:\\ \;\;\;\;\frac{t\_1 - \frac{t\_1}{\frac{\alpha}{\beta}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0)))
        (t_1 (/ (+ beta (+ beta 2.0)) alpha)))
   (if (<= t_0 -0.9998)
     (/ (- t_1 (/ t_1 (/ alpha beta))) 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 t_1 = (beta + (beta + 2.0)) / alpha;
	double tmp;
	if (t_0 <= -0.9998) {
		tmp = (t_1 - (t_1 / (alpha / beta))) / 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) :: t_1
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    t_1 = (beta + (beta + 2.0d0)) / alpha
    if (t_0 <= (-0.9998d0)) then
        tmp = (t_1 - (t_1 / (alpha / beta))) / 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 t_1 = (beta + (beta + 2.0)) / alpha;
	double tmp;
	if (t_0 <= -0.9998) {
		tmp = (t_1 - (t_1 / (alpha / beta))) / 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)
	t_1 = (beta + (beta + 2.0)) / alpha
	tmp = 0
	if t_0 <= -0.9998:
		tmp = (t_1 - (t_1 / (alpha / beta))) / 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))
	t_1 = Float64(Float64(beta + Float64(beta + 2.0)) / alpha)
	tmp = 0.0
	if (t_0 <= -0.9998)
		tmp = Float64(Float64(t_1 - Float64(t_1 / Float64(alpha / beta))) / 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);
	t_1 = (beta + (beta + 2.0)) / alpha;
	tmp = 0.0;
	if (t_0 <= -0.9998)
		tmp = (t_1 - (t_1 / (alpha / beta))) / 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]}, Block[{t$95$1 = N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9998], N[(N[(t$95$1 - N[(t$95$1 / N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $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) 2)) < -0.99980000000000002

   1. Initial program 7.6%

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

      1. flip3-+ 7.7%

         \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)}}}{2} \]

      2. add-exp-log 7.7%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)}\right)}}}{2} \]

      3. flip3-+ 7.6%

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

      4. +-commutative 7.6%

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

      5. log1p-udef 7.6%

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

      6. +-commutative 7.6%

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

      7. associate-+l+ 7.6%

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

   4. Applied egg-rr 7.6%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}}{2} \]
   5. Step-by-step derivation

      1. +-commutative 7.6%

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

      2. +-commutative 7.6%

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

      3. +-commutative 7.6%

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

   6. Simplified 7.6%

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

   7. Taylor expanded in alpha around inf 90.2%

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

      1. mul-1-neg 90.2%

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

      2. unsub-neg 90.2%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in beta around inf 99.3%

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

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha} - \frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{\frac{\alpha}{\beta}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9998)
     (/ (/ (+ 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 <= -0.9998) {
		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 <= (-0.9998d0)) 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 <= -0.9998) {
		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 <= -0.9998:
		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 <= -0.9998)
		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 <= -0.9998)
		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, -0.9998], 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) 2)) < -0.99980000000000002

   1. Initial program 7.6%

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

   3. Taylor expanded in alpha around -inf 98.6%

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

      1. associate-*r/ 98.6%

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

      2. sub-neg 98.6%

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

      3. mul-1-neg 98.6%

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

      4. distribute-lft-in 98.6%

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

      5. neg-mul-1 98.6%

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

      6. mul-1-neg 98.6%

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

      7. remove-double-neg 98.6%

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

      8. neg-mul-1 98.6%

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

      9. mul-1-neg 98.6%

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

      10. remove-double-neg 98.6%

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

   5. Simplified 98.6%

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

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 8.5e+19)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (+ (/ 1.0 alpha) (/ beta alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8.5e+19) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (1.0 / alpha) + (beta / 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 <= 8.5d+19) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (1.0d0 / alpha) + (beta / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8.5e+19) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 8.5e+19:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (1.0 / alpha) + (beta / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 8.5e+19)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 8.5e+19)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (1.0 / alpha) + (beta / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 8.5e+19], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.5e19

   1. Initial program 99.4%

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

   3. Taylor expanded in alpha around 0 97.3%

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

   if 8.5e19 < alpha

   1. Initial program 20.9%

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

   3. Taylor expanded in alpha around -inf 84.9%

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

      1. associate-*r/ 84.9%

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

      2. sub-neg 84.9%

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

      3. mul-1-neg 84.9%

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

      4. distribute-lft-in 84.9%

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

      5. neg-mul-1 84.9%

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

      6. mul-1-neg 84.9%

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

      7. remove-double-neg 84.9%

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

      8. neg-mul-1 84.9%

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

      9. mul-1-neg 84.9%

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

      10. remove-double-neg 84.9%

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

   5. Simplified 84.9%

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

   6. Taylor expanded in beta around 0 84.9%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 9.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 9.6e+19)
   (/ (+ 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 <= 9.6e+19) {
		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 <= 9.6d+19) 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 <= 9.6e+19) {
		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 <= 9.6e+19:
		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 <= 9.6e+19)
		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 <= 9.6e+19)
		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, 9.6e+19], 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 < 9.6e19

   1. Initial program 99.4%

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

   3. Taylor expanded in alpha around 0 97.3%

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

   if 9.6e19 < alpha

   1. Initial program 20.9%

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

   3. Taylor expanded in alpha around -inf 84.9%

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

      1. associate-*r/ 84.9%

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

      2. sub-neg 84.9%

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

      3. mul-1-neg 84.9%

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

      4. distribute-lft-in 84.9%

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

      5. neg-mul-1 84.9%

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

      6. mul-1-neg 84.9%

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

      7. remove-double-neg 84.9%

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

      8. neg-mul-1 84.9%

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

      9. mul-1-neg 84.9%

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

      10. remove-double-neg 84.9%

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

   5. Simplified 84.9%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.45 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.45e+21) 1.0 (+ (/ 1.0 alpha) (/ beta alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.45e+21) {
		tmp = 1.0;
	} else {
		tmp = (1.0 / alpha) + (beta / 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 <= 2.45d+21) then
        tmp = 1.0d0
    else
        tmp = (1.0d0 / alpha) + (beta / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.45e+21) {
		tmp = 1.0;
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.45e+21:
		tmp = 1.0
	else:
		tmp = (1.0 / alpha) + (beta / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.45e+21)
		tmp = 1.0;
	else
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.45e+21)
		tmp = 1.0;
	else
		tmp = (1.0 / alpha) + (beta / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.45e+21], 1.0, N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.45e21

   1. Initial program 99.4%

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

   3. Taylor expanded in beta around inf 43.4%

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

   if 2.45e21 < alpha

   1. Initial program 20.9%

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

   3. Taylor expanded in alpha around -inf 84.9%

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

      1. associate-*r/ 84.9%

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

      2. sub-neg 84.9%

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

      3. mul-1-neg 84.9%

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

      4. distribute-lft-in 84.9%

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

      5. neg-mul-1 84.9%

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

      6. mul-1-neg 84.9%

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

      7. remove-double-neg 84.9%

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

      8. neg-mul-1 84.9%

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

      9. mul-1-neg 84.9%

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

      10. remove-double-neg 84.9%

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

   5. Simplified 84.9%

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

   6. Taylor expanded in beta around 0 84.9%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.45 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = (1.0 + (beta * 0.5)) / 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 <= 2.0d0) then
        tmp = (1.0d0 + (beta * 0.5d0)) / 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 <= 2.0) {
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 74.5%

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

   3. Taylor expanded in alpha around 0 72.3%

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

   4. Taylor expanded in beta around 0 71.8%

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

      1. *-commutative 71.8%

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

   6. Simplified 71.8%

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

   if 2 < beta

   1. Initial program 85.0%

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

   3. Taylor expanded in beta around inf 81.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.75)
   (/ (- 1.0 (* alpha 0.5)) 2.0)
   (+ (/ 1.0 alpha) (/ beta alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.75) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else {
		tmp = (1.0 / alpha) + (beta / 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 <= 1.75d0) then
        tmp = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    else
        tmp = (1.0d0 / alpha) + (beta / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.75) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.75:
		tmp = (1.0 - (alpha * 0.5)) / 2.0
	else:
		tmp = (1.0 / alpha) + (beta / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.75)
		tmp = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0);
	else
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.75)
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	else
		tmp = (1.0 / alpha) + (beta / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.75], N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.75

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0 73.1%

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

      1. +-commutative 73.1%

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

   5. Simplified 73.1%

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

   6. Taylor expanded in alpha around 0 72.4%

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

      1. *-commutative 72.4%

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

   8. Simplified 72.4%

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

   if 1.75 < alpha

   1. Initial program 23.7%

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

   3. Taylor expanded in alpha around -inf 82.6%

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

      1. associate-*r/ 82.6%

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

      2. sub-neg 82.6%

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

      3. mul-1-neg 82.6%

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

      4. distribute-lft-in 82.6%

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

      5. neg-mul-1 82.6%

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

      6. mul-1-neg 82.6%

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

      7. remove-double-neg 82.6%

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

      8. neg-mul-1 82.6%

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

      9. mul-1-neg 82.6%

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

      10. remove-double-neg 82.6%

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

   5. Simplified 82.6%

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

   6. Taylor expanded in beta around 0 82.6%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1e+20) 1.0 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1e+20) {
		tmp = 1.0;
	} 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 <= 1d+20) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1e+20) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1e+20:
		tmp = 1.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1e+20)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1e+20)
		tmp = 1.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1e+20], 1.0, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1e20

   1. Initial program 99.4%

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

   3. Taylor expanded in beta around inf 43.4%

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

   if 1e20 < alpha

   1. Initial program 20.9%

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

   3. Taylor expanded in alpha around -inf 84.9%

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

      1. associate-*r/ 84.9%

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

      2. sub-neg 84.9%

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

      3. mul-1-neg 84.9%

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

      4. distribute-lft-in 84.9%

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

      5. neg-mul-1 84.9%

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

      6. mul-1-neg 84.9%

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

      7. remove-double-neg 84.9%

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

      8. neg-mul-1 84.9%

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

      9. mul-1-neg 84.9%

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

      10. remove-double-neg 84.9%

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

   5. Simplified 84.9%

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

   6. Taylor expanded in beta around 0 65.7%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 24.2% accurate, 4.3× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 (/ 1.0 alpha))

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0 / alpha

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in alpha around -inf 26.0%

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

   1. associate-*r/ 26.0%

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

   2. sub-neg 26.0%

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

   3. mul-1-neg 26.0%

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

   4. distribute-lft-in 26.0%

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

   5. neg-mul-1 26.0%

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

   6. mul-1-neg 26.0%

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

   7. remove-double-neg 26.0%

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

   8. neg-mul-1 26.0%

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

   9. mul-1-neg 26.0%

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

   10. remove-double-neg 26.0%

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

5. Simplified 26.0%

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

6. Taylor expanded in beta around 0 21.1%

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

7. Final simplification 21.1%

   \[\leadsto \frac{1}{\alpha} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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))