Octave 3.8, jcobi/1

Percentage Accurate: 74.7% → 99.8%

Time: 8.1s

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: 74.7% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999996:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{\left(\beta + \beta\right) - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} - \left(\frac{\alpha}{t_0} + -1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999999996)
     (/
      (+
       (* -2.0 (* (/ beta alpha) (/ beta 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)) <= -0.999999996) {
		tmp = ((-2.0 * ((beta / alpha) * (beta / alpha))) + (((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)) <= (-0.999999996d0)) then
        tmp = (((-2.0d0) * ((beta / alpha) * (beta / alpha))) + (((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)) <= -0.999999996) {
		tmp = ((-2.0 * ((beta / alpha) * (beta / alpha))) + (((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)) <= -0.999999996:
		tmp = ((-2.0 * ((beta / alpha) * (beta / 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)) <= -0.999999996)
		tmp = Float64(Float64(Float64(-2.0 * Float64(Float64(beta / alpha) * Float64(beta / alpha))) + Float64(Float64(Float64(beta + 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)) <= -0.999999996)
		tmp = ((-2.0 * ((beta / alpha) * (beta / alpha))) + (((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], -0.999999996], N[(N[(N[(-2.0 * N[(N[(beta / alpha), $MachinePrecision] * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(beta + beta), $MachinePrecision] - -2.0), $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)) < -0.999999996000000002

   1. Initial program 7.1%

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

      1. +-commutative 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in alpha around -inf 91.5%

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

      1. +-commutative 91.5%

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

      2. mul-1-neg 91.5%

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

      3. unsub-neg 91.5%

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

   6. Simplified 91.5%

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

   7. Taylor expanded in beta around inf 90.8%

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

      1. unpow2 90.8%

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

      2. unpow2 90.8%

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

      3. times-frac 99.2%

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

   9. Simplified 99.2%

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

   if -0.999999996000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. div-sub 99.7%

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

      2. associate-+l- 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+l+ 99.7%

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

   5. Applied egg-rr 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999996:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{\left(\beta + \beta\right) - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} + -1\right)}{2}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999996:\\ \;\;\;\;\frac{t_1 + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - \left(\frac{\alpha}{t_0} + -1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))) (t_1 (/ beta t_0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999999996)
     (/ (+ t_1 (/ (- beta -2.0) alpha)) 2.0)
     (/ (- t_1 (+ (/ alpha t_0) -1.0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999996) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_1 - ((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) :: t_1
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    t_1 = beta / t_0
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999999996d0)) then
        tmp = (t_1 + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = (t_1 - ((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 t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999996) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_1 - ((alpha / t_0) + -1.0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	t_1 = beta / t_0
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999996:
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = (t_1 - ((alpha / t_0) + -1.0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	t_1 = Float64(beta / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999999996)
		tmp = Float64(Float64(t_1 + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(t_1 - 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);
	t_1 = beta / t_0;
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999996)
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = (t_1 - ((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]}, Block[{t$95$1 = N[(beta / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999999996], N[(N[(t$95$1 + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$1 - 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)) < -0.999999996000000002

   1. Initial program 7.1%

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

      1. +-commutative 7.1%

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

   3. Simplified 7.1%

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

      1. div-sub 7.1%

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

      2. associate-+l- 11.0%

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

      3. associate-+l+ 11.0%

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

      4. associate-+l+ 11.0%

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

   5. Applied egg-rr 11.0%

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

   6. Taylor expanded in alpha around inf 99.0%

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

      1. associate-*r/ 99.0%

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

      2. +-commutative 99.0%

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

      3. distribute-lft-in 99.0%

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

      4. metadata-eval 99.0%

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

      5. neg-mul-1 99.0%

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

      6. unsub-neg 99.0%

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

   8. Simplified 99.0%

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

   if -0.999999996000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. div-sub 99.7%

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

      2. associate-+l- 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+l+ 99.7%

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

   5. Applied egg-rr 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999996:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} + -1\right)}{2}\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.999999996:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999999996)
     (/ (+ (/ beta (+ beta (+ alpha 2.0))) (/ (- 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 <= -0.999999996) {
		tmp = ((beta / (beta + (alpha + 2.0))) + ((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 <= (-0.999999996d0)) then
        tmp = ((beta / (beta + (alpha + 2.0d0))) + ((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 <= -0.999999996) {
		tmp = ((beta / (beta + (alpha + 2.0))) + ((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 <= -0.999999996:
		tmp = ((beta / (beta + (alpha + 2.0))) + ((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 <= -0.999999996)
		tmp = Float64(Float64(Float64(beta / Float64(beta + Float64(alpha + 2.0))) + Float64(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 <= -0.999999996)
		tmp = ((beta / (beta + (alpha + 2.0))) + ((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, -0.999999996], N[(N[(N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999999996000000002

   1. Initial program 7.1%

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

      1. +-commutative 7.1%

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

   3. Simplified 7.1%

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

      1. div-sub 7.1%

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

      2. associate-+l- 11.0%

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

      3. associate-+l+ 11.0%

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

      4. associate-+l+ 11.0%

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

   5. Applied egg-rr 11.0%

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

   6. Taylor expanded in alpha around inf 99.0%

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

      1. associate-*r/ 99.0%

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

      2. +-commutative 99.0%

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

      3. distribute-lft-in 99.0%

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

      4. metadata-eval 99.0%

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

      5. neg-mul-1 99.0%

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

      6. unsub-neg 99.0%

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

   8. Simplified 99.0%

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

   if -0.999999996000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999996:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.999999996:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999999996)
     (/ (/ (+ 2.0 (* 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 <= -0.999999996) {
		tmp = ((2.0 + (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 <= (-0.999999996d0)) then
        tmp = ((2.0d0 + (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 <= -0.999999996) {
		tmp = ((2.0 + (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 <= -0.999999996:
		tmp = ((2.0 + (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 <= -0.999999996)
		tmp = Float64(Float64(Float64(2.0 + 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 <= -0.999999996)
		tmp = ((2.0 + (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, -0.999999996], N[(N[(N[(2.0 + 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)) < -0.999999996000000002

   1. Initial program 7.1%

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

      1. +-commutative 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in alpha around inf 99.0%

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

   if -0.999999996000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999996:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 5: 68.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -1.26 \cdot 10^{-118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -1.75 \cdot 10^{-133}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq -4.8 \cdot 10^{-252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -5.8 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{-292}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{-243}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)))
   (if (<= beta -1.26e-118)
     t_0
     (if (<= beta -1.75e-133)
       (/ 1.0 alpha)
       (if (<= beta -4.8e-252)
         t_0
         (if (<= beta -5.8e-296)
           (/ 1.0 alpha)
           (if (<= beta 1.75e-292)
             t_0
             (if (<= beta 6.5e-243)
               (/ 1.0 alpha)
               (if (<= beta 2.0) t_0 1.0)))))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= -1.26e-118) {
		tmp = t_0;
	} else if (beta <= -1.75e-133) {
		tmp = 1.0 / alpha;
	} else if (beta <= -4.8e-252) {
		tmp = t_0;
	} else if (beta <= -5.8e-296) {
		tmp = 1.0 / alpha;
	} else if (beta <= 1.75e-292) {
		tmp = t_0;
	} else if (beta <= 6.5e-243) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    if (beta <= (-1.26d-118)) then
        tmp = t_0
    else if (beta <= (-1.75d-133)) then
        tmp = 1.0d0 / alpha
    else if (beta <= (-4.8d-252)) then
        tmp = t_0
    else if (beta <= (-5.8d-296)) then
        tmp = 1.0d0 / alpha
    else if (beta <= 1.75d-292) then
        tmp = t_0
    else if (beta <= 6.5d-243) then
        tmp = 1.0d0 / alpha
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= -1.26e-118) {
		tmp = t_0;
	} else if (beta <= -1.75e-133) {
		tmp = 1.0 / alpha;
	} else if (beta <= -4.8e-252) {
		tmp = t_0;
	} else if (beta <= -5.8e-296) {
		tmp = 1.0 / alpha;
	} else if (beta <= 1.75e-292) {
		tmp = t_0;
	} else if (beta <= 6.5e-243) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (beta * 0.5)) / 2.0
	tmp = 0
	if beta <= -1.26e-118:
		tmp = t_0
	elif beta <= -1.75e-133:
		tmp = 1.0 / alpha
	elif beta <= -4.8e-252:
		tmp = t_0
	elif beta <= -5.8e-296:
		tmp = 1.0 / alpha
	elif beta <= 1.75e-292:
		tmp = t_0
	elif beta <= 6.5e-243:
		tmp = 1.0 / alpha
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0)
	tmp = 0.0
	if (beta <= -1.26e-118)
		tmp = t_0;
	elseif (beta <= -1.75e-133)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= -4.8e-252)
		tmp = t_0;
	elseif (beta <= -5.8e-296)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 1.75e-292)
		tmp = t_0;
	elseif (beta <= 6.5e-243)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta * 0.5)) / 2.0;
	tmp = 0.0;
	if (beta <= -1.26e-118)
		tmp = t_0;
	elseif (beta <= -1.75e-133)
		tmp = 1.0 / alpha;
	elseif (beta <= -4.8e-252)
		tmp = t_0;
	elseif (beta <= -5.8e-296)
		tmp = 1.0 / alpha;
	elseif (beta <= 1.75e-292)
		tmp = t_0;
	elseif (beta <= 6.5e-243)
		tmp = 1.0 / alpha;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, -1.26e-118], t$95$0, If[LessEqual[beta, -1.75e-133], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, -4.8e-252], t$95$0, If[LessEqual[beta, -5.8e-296], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 1.75e-292], t$95$0, If[LessEqual[beta, 6.5e-243], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 2.0], t$95$0, 1.0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < -1.26e-118 or -1.75000000000000001e-133 < beta < -4.8000000000000003e-252 or -5.79999999999999965e-296 < beta < 1.75e-292 or 6.50000000000000043e-243 < beta < 2

   1. Initial program 78.8%

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

      1. +-commutative 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in beta around 0 78.1%

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

      1. +-commutative 78.1%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in alpha around 0 75.3%

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

      1. *-commutative 75.3%

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

   9. Simplified 75.3%

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

   if -1.26e-118 < beta < -1.75000000000000001e-133 or -4.8000000000000003e-252 < beta < -5.79999999999999965e-296 or 1.75e-292 < beta < 6.50000000000000043e-243

   1. Initial program 36.2%

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

      1. +-commutative 36.2%

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

   3. Simplified 36.2%

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

   4. Taylor expanded in beta around 0 36.2%

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

      1. +-commutative 36.2%

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

   6. Simplified 36.2%

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

   7. Taylor expanded in alpha around inf 69.7%

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

   8. Taylor expanded in beta around 0 69.7%

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

   if 2 < beta

   1. Initial program 82.1%

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

      1. +-commutative 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in beta around inf 79.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -1.26 \cdot 10^{-118}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq -1.75 \cdot 10^{-133}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq -4.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq -5.8 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{-292}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{-243}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.6:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.6 \cdot 10^{+197} \lor \neg \left(\alpha \leq 1.65 \cdot 10^{+230}\right):\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.6)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (or (<= alpha 2.6e+197) (not (<= alpha 1.65e+230)))
     (/ (/ (+ beta 2.0) alpha) 2.0)
     (/ (* 2.0 (/ beta alpha)) 2.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.6) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 2.6e+197) || !(alpha <= 1.65e+230)) {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	} else {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.6d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if ((alpha <= 2.6d+197) .or. (.not. (alpha <= 1.65d+230))) then
        tmp = ((beta + 2.0d0) / alpha) / 2.0d0
    else
        tmp = (2.0d0 * (beta / alpha)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.6) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 2.6e+197) || !(alpha <= 1.65e+230)) {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	} else {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.6:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif (alpha <= 2.6e+197) or not (alpha <= 1.65e+230):
		tmp = ((beta + 2.0) / alpha) / 2.0
	else:
		tmp = (2.0 * (beta / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.6)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif ((alpha <= 2.6e+197) || !(alpha <= 1.65e+230))
		tmp = Float64(Float64(Float64(beta + 2.0) / alpha) / 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(beta / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.6)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif ((alpha <= 2.6e+197) || ~((alpha <= 1.65e+230)))
		tmp = ((beta + 2.0) / alpha) / 2.0;
	else
		tmp = (2.0 * (beta / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.6], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 2.6e+197], N[Not[LessEqual[alpha, 1.65e+230]], $MachinePrecision]], N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 2.60000000000000009

   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 98.2%

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

   if 2.60000000000000009 < alpha < 2.59999999999999987e197 or 1.65000000000000007e230 < alpha

   1. Initial program 32.6%

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

      1. +-commutative 32.6%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in beta around 0 9.7%

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

      1. +-commutative 9.7%

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

   6. Simplified 9.7%

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

   7. Taylor expanded in alpha around inf 66.7%

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

   if 2.59999999999999987e197 < alpha < 1.65000000000000007e230

   1. Initial program 11.7%

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

      1. +-commutative 11.7%

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

   3. Simplified 11.7%

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

      1. div-sub 11.7%

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

      2. associate-+l- 20.8%

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

      3. associate-+l+ 20.8%

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

      4. associate-+l+ 20.8%

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

   5. Applied egg-rr 20.8%

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

   6. Taylor expanded in alpha around inf 93.9%

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

      1. *-commutative 93.9%

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

   8. Simplified 93.9%

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

   9. Taylor expanded in beta around inf 63.6%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.6:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.6 \cdot 10^{+197} \lor \neg \left(\alpha \leq 1.65 \cdot 10^{+230}\right):\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 93.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.6:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.6)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.6) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.6d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.6) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.6:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.6)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.6)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.6], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.60000000000000009

   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 98.2%

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

   if 2.60000000000000009 < alpha

   1. Initial program 28.6%

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

      1. +-commutative 28.6%

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

   3. Simplified 28.6%

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

   4. Taylor expanded in alpha around inf 78.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.6:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 50.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.08 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.08e-11) (/ 1.0 alpha) 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.08e-11) {
		tmp = 1.0 / alpha;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.08d-11) then
        tmp = 1.0d0 / alpha
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.08e-11) {
		tmp = 1.0 / alpha;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.08e-11:
		tmp = 1.0 / alpha
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.08e-11)
		tmp = Float64(1.0 / alpha);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.08e-11)
		tmp = 1.0 / alpha;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.08e-11], N[(1.0 / alpha), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.07999999999999992e-11

   1. Initial program 69.1%

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

      1. +-commutative 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in beta around 0 68.6%

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

      1. +-commutative 68.6%

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

   6. Simplified 68.6%

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

   7. Taylor expanded in alpha around inf 35.8%

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

   8. Taylor expanded in beta around 0 35.7%

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

   if 1.07999999999999992e-11 < beta

   1. Initial program 81.9%

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

      1. +-commutative 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in beta around inf 77.3%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.08 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 24.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. +-commutative 74.0%

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

3. Simplified 74.0%

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

4. Taylor expanded in beta around 0 45.3%

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

   1. +-commutative 45.3%

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

6. Simplified 45.3%

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

7. Taylor expanded in alpha around inf 24.4%

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

8. Taylor expanded in beta around 0 23.7%

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

9. Final simplification 23.7%

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

Reproduce

herbie shell --seed 2023181 
(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))