Octave 3.8, jcobi/1

Percentage Accurate: 75.6% → 99.7%

Time: 11.0s

Alternatives: 11

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.6% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.96:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.96)
   (/
    (/
     (+
      2.0
      (-
       (* beta (+ 2.0 (- (* (/ beta alpha) -2.0) (/ 6.0 alpha))))
       (/ 4.0 alpha)))
     alpha)
    2.0)
   (/ (+ (* (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0)))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96) {
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0;
	} else {
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 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) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.96d0)) then
        tmp = ((2.0d0 + ((beta * (2.0d0 + (((beta / alpha) * (-2.0d0)) - (6.0d0 / alpha)))) - (4.0d0 / alpha))) / alpha) / 2.0d0
    else
        tmp = (((beta - alpha) * (1.0d0 / (beta + (alpha + 2.0d0)))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96) {
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0;
	} else {
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96:
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0
	else:
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.96)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * Float64(2.0 + Float64(Float64(Float64(beta / alpha) * -2.0) - Float64(6.0 / alpha)))) - Float64(4.0 / alpha))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) * Float64(1.0 / Float64(beta + Float64(alpha + 2.0)))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96)
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0;
	else
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.96], N[(N[(N[(2.0 + N[(N[(beta * N[(2.0 + N[(N[(N[(beta / alpha), $MachinePrecision] * -2.0), $MachinePrecision] - N[(6.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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))) < -0.95999999999999996

   1. Initial program 7.4%

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

      1. +-commutative 7.4%

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

   3. Simplified 7.4%

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

   5. Taylor expanded in alpha around inf 98.7%

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

      1. +-commutative 98.7%

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

      2. associate--l+ 98.7%

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

      3. +-commutative 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. sub-neg 98.7%

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

      7. associate-/l* 98.7%

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

      8. distribute-rgt-neg-in 98.7%

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

      9. distribute-frac-neg 98.7%

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

      10. distribute-neg-in 98.7%

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

      11. metadata-eval 98.7%

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

      12. unsub-neg 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in beta around 0 100.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(2 + \beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) - 6 \cdot \frac{1}{\alpha}\right)\right) - 4 \cdot \frac{1}{\alpha}}}{\alpha}}{2} \]
   9. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \frac{\frac{\color{blue}{2 + \left(\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) - 6 \cdot \frac{1}{\alpha}\right) - 4 \cdot \frac{1}{\alpha}\right)}}{\alpha}}{2} \]

      2. associate--l+ 100.0%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \color{blue}{\left(2 + \left(-2 \cdot \frac{\beta}{\alpha} - 6 \cdot \frac{1}{\alpha}\right)\right)} - 4 \cdot \frac{1}{\alpha}\right)}{\alpha}}{2} \]

      3. *-commutative 100.0%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\color{blue}{\frac{\beta}{\alpha} \cdot -2} - 6 \cdot \frac{1}{\alpha}\right)\right) - 4 \cdot \frac{1}{\alpha}\right)}{\alpha}}{2} \]

      4. associate-*r/ 100.0%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \color{blue}{\frac{6 \cdot 1}{\alpha}}\right)\right) - 4 \cdot \frac{1}{\alpha}\right)}{\alpha}}{2} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{\color{blue}{6}}{\alpha}\right)\right) - 4 \cdot \frac{1}{\alpha}\right)}{\alpha}}{2} \]

      6. associate-*r/ 100.0%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \color{blue}{\frac{4 \cdot 1}{\alpha}}\right)}{\alpha}}{2} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{\color{blue}{4}}{\alpha}\right)}{\alpha}}{2} \]

   10. Simplified 100.0%

       \[\leadsto \frac{\frac{\color{blue}{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}}{\alpha}}{2} \]

   if -0.95999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.96:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.96:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(\beta - -2\right)\right) - \left(\beta + 2\right) \cdot \frac{\beta + \left(\beta + 2\right)}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.96)
   (/
    (/
     (-
      (+ beta (- beta -2.0))
      (* (+ beta 2.0) (/ (+ beta (+ beta 2.0)) alpha)))
     alpha)
    2.0)
   (/ (+ (* (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0)))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96) {
		tmp = (((beta + (beta - -2.0)) - ((beta + 2.0) * ((beta + (beta + 2.0)) / alpha))) / alpha) / 2.0;
	} else {
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 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) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.96d0)) then
        tmp = (((beta + (beta - (-2.0d0))) - ((beta + 2.0d0) * ((beta + (beta + 2.0d0)) / alpha))) / alpha) / 2.0d0
    else
        tmp = (((beta - alpha) * (1.0d0 / (beta + (alpha + 2.0d0)))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96) {
		tmp = (((beta + (beta - -2.0)) - ((beta + 2.0) * ((beta + (beta + 2.0)) / alpha))) / alpha) / 2.0;
	} else {
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96:
		tmp = (((beta + (beta - -2.0)) - ((beta + 2.0) * ((beta + (beta + 2.0)) / alpha))) / alpha) / 2.0
	else:
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.96)
		tmp = Float64(Float64(Float64(Float64(beta + Float64(beta - -2.0)) - Float64(Float64(beta + 2.0) * Float64(Float64(beta + Float64(beta + 2.0)) / alpha))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) * Float64(1.0 / Float64(beta + Float64(alpha + 2.0)))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96)
		tmp = (((beta + (beta - -2.0)) - ((beta + 2.0) * ((beta + (beta + 2.0)) / alpha))) / alpha) / 2.0;
	else
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.96], N[(N[(N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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))) < -0.95999999999999996

   1. Initial program 7.4%

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

      1. +-commutative 7.4%

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

   3. Simplified 7.4%

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

      1. clear-num 7.4%

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

      2. associate-/r/ 7.4%

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

      3. associate-+l+ 7.4%

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

   6. Applied egg-rr 7.4%

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

   7. Taylor expanded in alpha around -inf 98.7%

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

   9. Simplified 98.7%

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

       1. distribute-frac-neg2 98.7%

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

       2. associate-+l- 98.7%

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

       3. associate-/l* 100.0%

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

   11. Applied egg-rr 100.0%

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

   if -0.95999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.96:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(\beta - -2\right)\right) - \left(\beta + 2\right) \cdot \frac{\beta + \left(\beta + 2\right)}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.96:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 2 - \frac{4}{\alpha}\right) + \left(2 - \beta \cdot \frac{\beta}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.96)
   (/
    (/
     (+ (- (* beta 2.0) (/ 4.0 alpha)) (- 2.0 (* beta (/ beta alpha))))
     alpha)
    2.0)
   (/ (+ (* (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0)))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96) {
		tmp = ((((beta * 2.0) - (4.0 / alpha)) + (2.0 - (beta * (beta / alpha)))) / alpha) / 2.0;
	} else {
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 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) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.96d0)) then
        tmp = ((((beta * 2.0d0) - (4.0d0 / alpha)) + (2.0d0 - (beta * (beta / alpha)))) / alpha) / 2.0d0
    else
        tmp = (((beta - alpha) * (1.0d0 / (beta + (alpha + 2.0d0)))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96) {
		tmp = ((((beta * 2.0) - (4.0 / alpha)) + (2.0 - (beta * (beta / alpha)))) / alpha) / 2.0;
	} else {
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96:
		tmp = ((((beta * 2.0) - (4.0 / alpha)) + (2.0 - (beta * (beta / alpha)))) / alpha) / 2.0
	else:
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.96)
		tmp = Float64(Float64(Float64(Float64(Float64(beta * 2.0) - Float64(4.0 / alpha)) + Float64(2.0 - Float64(beta * Float64(beta / alpha)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) * Float64(1.0 / Float64(beta + Float64(alpha + 2.0)))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96)
		tmp = ((((beta * 2.0) - (4.0 / alpha)) + (2.0 - (beta * (beta / alpha)))) / alpha) / 2.0;
	else
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.96], N[(N[(N[(N[(N[(beta * 2.0), $MachinePrecision] - N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 - N[(beta * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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))) < -0.95999999999999996

   1. Initial program 7.4%

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

      1. +-commutative 7.4%

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

   3. Simplified 7.4%

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

   5. Taylor expanded in alpha around inf 98.7%

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

      1. +-commutative 98.7%

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

      2. associate--l+ 98.7%

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

      3. +-commutative 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. sub-neg 98.7%

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

      7. associate-/l* 98.7%

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

      8. distribute-rgt-neg-in 98.7%

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

      9. distribute-frac-neg 98.7%

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

      10. distribute-neg-in 98.7%

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

      11. metadata-eval 98.7%

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

      12. unsub-neg 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in beta around 0 99.6%

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

   9. Taylor expanded in beta around inf 99.6%

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

       1. neg-mul-1 99.6%

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

       2. distribute-neg-frac 99.6%

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

   11. Simplified 99.6%

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

   if -0.95999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.96:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 2 - \frac{4}{\alpha}\right) + \left(2 - \beta \cdot \frac{\beta}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.96:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.96)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
   (/ (+ (* (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0)))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 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) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.96d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (((beta - alpha) * (1.0d0 / (beta + (alpha + 2.0d0)))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.96)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) * Float64(1.0 / Float64(beta + Float64(alpha + 2.0)))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.96)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = (((beta - alpha) * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.96], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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))) < -0.95999999999999996

   1. Initial program 7.4%

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

      1. +-commutative 7.4%

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

   3. Simplified 7.4%

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

   5. Taylor expanded in alpha around -inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. sub-neg 99.2%

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

      3. distribute-lft-in 99.2%

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

      4. neg-mul-1 99.2%

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

      5. mul-1-neg 99.2%

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

      6. remove-double-neg 99.2%

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

      7. neg-mul-1 99.2%

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

      8. remove-double-neg 99.2%

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

   7. Simplified 99.2%

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

   if -0.95999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.96:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.96:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.96)
     (/ (/ (+ 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 <= -0.96) {
		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 <= (-0.96d0)) 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 <= -0.96) {
		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 <= -0.96:
		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 <= -0.96)
		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 <= -0.96)
		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, -0.96], 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))) < -0.95999999999999996

   1. Initial program 7.4%

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

      1. +-commutative 7.4%

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

   3. Simplified 7.4%

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

   5. Taylor expanded in alpha around -inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. sub-neg 99.2%

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

      3. distribute-lft-in 99.2%

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

      4. neg-mul-1 99.2%

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

      5. mul-1-neg 99.2%

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

      6. remove-double-neg 99.2%

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

      7. neg-mul-1 99.2%

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

      8. remove-double-neg 99.2%

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

   7. Simplified 99.2%

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

   if -0.95999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.96:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3800000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3800000000.0)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3800000000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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 <= 3800000000.0d0) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 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 <= 3800000000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 3800000000.0:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3800000000.0)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 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 <= 3800000000.0)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 3800000000.0], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $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 < 3.8e9

   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 3.8e9 < alpha

   1. Initial program 18.0%

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

      1. +-commutative 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in alpha around -inf 88.6%

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

      1. associate-*r/ 88.6%

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

      2. sub-neg 88.6%

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

      3. distribute-lft-in 88.6%

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

      4. neg-mul-1 88.6%

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

      5. mul-1-neg 88.6%

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

      6. remove-double-neg 88.6%

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

      7. neg-mul-1 88.6%

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

      8. remove-double-neg 88.6%

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

   7. Simplified 88.6%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3800000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8200000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 8200000000.0)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8200000000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 8200000000.0d0) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8200000000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 8200000000.0:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 8200000000.0)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 8200000000.0)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 8200000000.0], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.2e9

   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 8.2e9 < alpha

   1. Initial program 18.0%

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

      1. +-commutative 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in alpha around -inf 88.6%

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

      1. associate-*r/ 88.6%

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

      2. sub-neg 88.6%

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

      3. distribute-lft-in 88.6%

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

      4. neg-mul-1 88.6%

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

      5. mul-1-neg 88.6%

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

      6. remove-double-neg 88.6%

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

      7. neg-mul-1 88.6%

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

      8. remove-double-neg 88.6%

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

   7. Simplified 88.6%

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

   8. Taylor expanded in beta around 0 69.8%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8200000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.85:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.85) (/ (- 1.0 (* alpha 0.5)) 2.0) (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.85) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 0.85d0) then
        tmp = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.85) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.85:
		tmp = (1.0 - (alpha * 0.5)) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.85)
		tmp = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0);
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.85)
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.85], N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.849999999999999978

   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 beta around 0 73.4%

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

      1. +-commutative 73.4%

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

   7. Simplified 73.4%

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

   8. Taylor expanded in alpha around 0 72.6%

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

      1. *-commutative 72.6%

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

   10. Simplified 72.6%

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

   if 0.849999999999999978 < alpha

   1. Initial program 20.7%

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

      1. +-commutative 20.7%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in alpha around -inf 85.9%

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

      1. associate-*r/ 85.9%

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

      2. sub-neg 85.9%

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

      3. distribute-lft-in 85.9%

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

      4. neg-mul-1 85.9%

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

      5. mul-1-neg 85.9%

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

      6. remove-double-neg 85.9%

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

      7. neg-mul-1 85.9%

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

      8. remove-double-neg 85.9%

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

   7. Simplified 85.9%

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

   8. Taylor expanded in beta around 0 67.9%

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

Alternative 9: 66.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.33:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.33) (/ (+ 1.0 (* beta 0.5)) 2.0) (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.33) {
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 0.33d0) then
        tmp = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.33) {
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.33:
		tmp = (1.0 + (beta * 0.5)) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.33)
		tmp = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0);
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.33)
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.33], N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.330000000000000016

   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 69.0%

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

      1. *-commutative 69.0%

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

   8. Simplified 69.0%

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

   if 0.330000000000000016 < alpha

   1. Initial program 20.7%

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

      1. +-commutative 20.7%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in alpha around -inf 85.9%

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

      1. associate-*r/ 85.9%

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

      2. sub-neg 85.9%

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

      3. distribute-lft-in 85.9%

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

      4. neg-mul-1 85.9%

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

      5. mul-1-neg 85.9%

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

      6. remove-double-neg 85.9%

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

      7. neg-mul-1 85.9%

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

      8. remove-double-neg 85.9%

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

   7. Simplified 85.9%

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

   8. Taylor expanded in beta around 0 67.9%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 0.33:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2700000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2700000000.0) 1.0 (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2700000000.0) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2700000000.0d0) then
        tmp = 1.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2700000000.0) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2700000000.0:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2700000000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2700000000.0)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2700000000.0], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.7e9

   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 beta around inf 43.7%

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

   if 2.7e9 < alpha

   1. Initial program 18.0%

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

      1. +-commutative 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in alpha around -inf 88.6%

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

      1. associate-*r/ 88.6%

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

      2. sub-neg 88.6%

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

      3. distribute-lft-in 88.6%

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

      4. neg-mul-1 88.6%

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

      5. mul-1-neg 88.6%

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

      6. remove-double-neg 88.6%

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

      7. neg-mul-1 88.6%

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

      8. remove-double-neg 88.6%

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

   7. Simplified 88.6%

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

   8. Taylor expanded in beta around 0 69.8%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2700000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.4% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 72.1%

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

   1. +-commutative 72.1%

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

3. Simplified 72.1%

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

5. Taylor expanded in beta around inf 34.0%

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

6. Final simplification 34.0%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024087 
(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))