Octave 3.8, jcobi/1

Percentage Accurate: 75.8% → 99.9%

Time: 7.8s

Alternatives: 9

Speedup: 1.2×

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: 75.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} - \left(\frac{\alpha}{t_0} + -1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9995)
     (/
      (fma
       (/ (- (- -2.0 beta) beta) alpha)
       (/ (+ beta 2.0) alpha)
       (/ (+ beta (+ beta 2.0)) alpha))
      2.0)
     (/ (- (/ beta t_0) (+ (/ alpha t_0) -1.0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995) {
		tmp = fma((((-2.0 - beta) - beta) / alpha), ((beta + 2.0) / alpha), ((beta + (beta + 2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((beta / t_0) - ((alpha / t_0) + -1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9995)
		tmp = Float64(fma(Float64(Float64(Float64(-2.0 - beta) - beta) / alpha), Float64(Float64(beta + 2.0) / alpha), Float64(Float64(beta + Float64(beta + 2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) - Float64(Float64(alpha / t_0) + -1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9995], N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] - N[(N[(alpha / t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99950000000000006

   1. Initial program 6.8%

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

      1. +-commutative 6.8%

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

   3. Simplified 6.8%

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

   4. Taylor expanded in alpha around -inf 96.2%

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

      1. +-commutative 96.2%

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

      2. mul-1-neg 96.2%

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

      3. unsub-neg 96.2%

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

   6. Simplified 100.0%

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

   if -0.99950000000000006 < (/.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. Step-by-step derivation

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} + -1\right)}{2}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} - \left(\frac{\alpha}{t_0} + -1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9995)
     (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
     (/ (- (/ beta t_0) (+ (/ alpha t_0) -1.0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) - ((alpha / 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 + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9995d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((beta / t_0) - ((alpha / 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 + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) - ((alpha / t_0) + -1.0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = ((beta / t_0) - ((alpha / t_0) + -1.0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9995)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) - Float64(Float64(alpha / t_0) + -1.0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = ((beta / t_0) - ((alpha / t_0) + -1.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9995], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] - N[(N[(alpha / t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99950000000000006

   1. Initial program 6.8%

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

      1. +-commutative 6.8%

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

   3. Simplified 6.8%

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

   4. Taylor expanded in alpha around -inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. sub-neg 99.7%

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

      3. mul-1-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. neg-mul-1 99.7%

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

      6. mul-1-neg 99.7%

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

      7. remove-double-neg 99.7%

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

      8. neg-mul-1 99.7%

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

      9. mul-1-neg 99.7%

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

      10. remove-double-neg 99.7%

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

      11. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   if -0.99950000000000006 < (/.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. Step-by-step derivation

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} + -1\right)}{2}\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9995:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9995)
     (/ (/ (+ 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 <= -0.9995) {
		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 <= (-0.9995d0)) 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 <= -0.9995) {
		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 <= -0.9995:
		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 <= -0.9995)
		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 <= -0.9995)
		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, -0.9995], 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)) < -0.99950000000000006

   1. Initial program 6.8%

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

      1. +-commutative 6.8%

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

   3. Simplified 6.8%

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

   4. Taylor expanded in alpha around -inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. sub-neg 99.7%

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

      3. mul-1-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. neg-mul-1 99.7%

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

      6. mul-1-neg 99.7%

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

      7. remove-double-neg 99.7%

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

      8. neg-mul-1 99.7%

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

      9. mul-1-neg 99.7%

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

      10. remove-double-neg 99.7%

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

      11. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   if -0.99950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 4: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{if}\;\alpha \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.25:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (* alpha 0.5)) 2.0)))
   (if (<= alpha 7.5e-116)
     t_0
     (if (<= alpha 3.8e-112)
       1.0
       (if (<= alpha 1.25) t_0 (/ (/ 2.0 alpha) 2.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= 7.5e-116) {
		tmp = t_0;
	} else if (alpha <= 3.8e-112) {
		tmp = 1.0;
	} else if (alpha <= 1.25) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    if (alpha <= 7.5d-116) then
        tmp = t_0
    else if (alpha <= 3.8d-112) then
        tmp = 1.0d0
    else if (alpha <= 1.25d0) then
        tmp = t_0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= 7.5e-116) {
		tmp = t_0;
	} else if (alpha <= 3.8e-112) {
		tmp = 1.0;
	} else if (alpha <= 1.25) {
		tmp = t_0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 - (alpha * 0.5)) / 2.0
	tmp = 0
	if alpha <= 7.5e-116:
		tmp = t_0
	elif alpha <= 3.8e-112:
		tmp = 1.0
	elif alpha <= 1.25:
		tmp = t_0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0)
	tmp = 0.0
	if (alpha <= 7.5e-116)
		tmp = t_0;
	elseif (alpha <= 3.8e-112)
		tmp = 1.0;
	elseif (alpha <= 1.25)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	tmp = 0.0;
	if (alpha <= 7.5e-116)
		tmp = t_0;
	elseif (alpha <= 3.8e-112)
		tmp = 1.0;
	elseif (alpha <= 1.25)
		tmp = t_0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 7.5e-116], t$95$0, If[LessEqual[alpha, 3.8e-112], 1.0, If[LessEqual[alpha, 1.25], t$95$0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 7.5000000000000004e-116 or 3.79999999999999995e-112 < alpha < 1.25

   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. Taylor expanded in beta around 0 77.1%

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

      1. +-commutative 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in alpha around 0 76.5%

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

      1. *-commutative 76.5%

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

   9. Simplified 76.5%

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

   if 7.5000000000000004e-116 < alpha < 3.79999999999999995e-112

   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. Taylor expanded in beta around inf 100.0%

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

   if 1.25 < alpha

   1. Initial program 20.6%

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

      1. +-commutative 20.6%

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

   3. Simplified 20.6%

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

   4. Taylor expanded in alpha around -inf 86.6%

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

      1. associate-*r/ 86.6%

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

      2. sub-neg 86.6%

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

      3. mul-1-neg 86.6%

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

      4. distribute-lft-in 86.6%

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

      5. neg-mul-1 86.6%

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

      6. mul-1-neg 86.6%

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

      7. remove-double-neg 86.6%

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

      8. neg-mul-1 86.6%

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

      9. mul-1-neg 86.6%

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

      10. remove-double-neg 86.6%

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

      11. +-commutative 86.6%

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

   6. Simplified 86.6%

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

   7. Taylor expanded in beta around 0 69.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.25:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 69.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.2 \cdot 10^{-116}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 5600000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 5.2e-116)
   0.5
   (if (<= alpha 3.8e-112)
     1.0
     (if (<= alpha 5600000.0) 0.5 (/ (/ 2.0 alpha) 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5.2e-116) {
		tmp = 0.5;
	} else if (alpha <= 3.8e-112) {
		tmp = 1.0;
	} else if (alpha <= 5600000.0) {
		tmp = 0.5;
	} 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 <= 5.2d-116) then
        tmp = 0.5d0
    else if (alpha <= 3.8d-112) then
        tmp = 1.0d0
    else if (alpha <= 5600000.0d0) then
        tmp = 0.5d0
    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 <= 5.2e-116) {
		tmp = 0.5;
	} else if (alpha <= 3.8e-112) {
		tmp = 1.0;
	} else if (alpha <= 5600000.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 5.2e-116:
		tmp = 0.5
	elif alpha <= 3.8e-112:
		tmp = 1.0
	elif alpha <= 5600000.0:
		tmp = 0.5
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 5.2e-116)
		tmp = 0.5;
	elseif (alpha <= 3.8e-112)
		tmp = 1.0;
	elseif (alpha <= 5600000.0)
		tmp = 0.5;
	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 <= 5.2e-116)
		tmp = 0.5;
	elseif (alpha <= 3.8e-112)
		tmp = 1.0;
	elseif (alpha <= 5600000.0)
		tmp = 0.5;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 5.2e-116], 0.5, If[LessEqual[alpha, 3.8e-112], 1.0, If[LessEqual[alpha, 5600000.0], 0.5, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 5.2000000000000001e-116 or 3.79999999999999995e-112 < alpha < 5.6e6

   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. Step-by-step derivation

      1. div-sub 100.0%

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

      2. associate-+l- 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+l+ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in alpha around 0 98.0%

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

   7. Taylor expanded in beta around 0 75.1%

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

   if 5.2000000000000001e-116 < alpha < 3.79999999999999995e-112

   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. Taylor expanded in beta around inf 100.0%

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

   if 5.6e6 < 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. Taylor expanded in alpha around -inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. sub-neg 88.9%

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

      3. mul-1-neg 88.9%

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

      4. distribute-lft-in 88.9%

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

      5. neg-mul-1 88.9%

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

      6. mul-1-neg 88.9%

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

      7. remove-double-neg 88.9%

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

      8. neg-mul-1 88.9%

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

      9. mul-1-neg 88.9%

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

      10. remove-double-neg 88.9%

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

      11. +-commutative 88.9%

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

   6. Simplified 88.9%

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

   7. Taylor expanded in beta around 0 70.8%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.2 \cdot 10^{-116}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 5600000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 88.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 122000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 122000000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 122000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 122000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 122000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 122000000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 122000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 122000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 122000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.22e8

   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. Taylor expanded in alpha around 0 98.1%

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

   if 1.22e8 < 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. Taylor expanded in alpha around -inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. sub-neg 88.9%

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

      3. mul-1-neg 88.9%

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

      4. distribute-lft-in 88.9%

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

      5. neg-mul-1 88.9%

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

      6. mul-1-neg 88.9%

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

      7. remove-double-neg 88.9%

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

      8. neg-mul-1 88.9%

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

      9. mul-1-neg 88.9%

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

      10. remove-double-neg 88.9%

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

      11. +-commutative 88.9%

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

   6. Simplified 88.9%

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

   7. Taylor expanded in beta around 0 70.8%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 122000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 93.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 340000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 340000000.0)
   (/ (+ 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 <= 340000000.0) {
		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 <= 340000000.0d0) 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 <= 340000000.0) {
		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 <= 340000000.0:
		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 <= 340000000.0)
		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 <= 340000000.0)
		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, 340000000.0], 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 < 3.4e8

   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. Taylor expanded in alpha around 0 98.1%

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

   if 3.4e8 < 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. Taylor expanded in alpha around -inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. sub-neg 88.9%

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

      3. mul-1-neg 88.9%

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

      4. distribute-lft-in 88.9%

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

      5. neg-mul-1 88.9%

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

      6. mul-1-neg 88.9%

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

      7. remove-double-neg 88.9%

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

      8. neg-mul-1 88.9%

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

      9. mul-1-neg 88.9%

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

      10. remove-double-neg 88.9%

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

      11. +-commutative 88.9%

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

   6. Simplified 88.9%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 340000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 72.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 2.1) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.1d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.1)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.1], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 67.4%

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

      1. +-commutative 67.4%

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

   3. Simplified 67.4%

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

      1. div-sub 67.4%

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

      2. associate-+l- 67.5%

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

      3. associate-+l+ 67.5%

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

      4. associate-+l+ 67.5%

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

   5. Applied egg-rr 67.5%

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

   6. Taylor expanded in alpha around 0 64.9%

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

   7. Taylor expanded in beta around 0 64.6%

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

   if 2.10000000000000009 < beta

   1. Initial program 79.8%

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

      1. +-commutative 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in beta around inf 73.9%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 50.7% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 71.2%

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

   1. +-commutative 71.2%

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

3. Simplified 71.2%

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

   1. div-sub 71.2%

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

   2. associate-+l- 72.1%

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

   3. associate-+l+ 72.1%

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

   4. associate-+l+ 72.1%

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

5. Applied egg-rr 72.1%

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

6. Taylor expanded in alpha around 0 68.4%

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

7. Taylor expanded in beta around 0 50.0%

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

8. Final simplification 50.0%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023171 
(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))