Octave 3.8, jcobi/1

Percentage Accurate: 74.2% → 99.7%

Time: 50.9s

Alternatives: 13

Speedup: 1.3×

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: 74.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 9.5%

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

      1. +-commutative 9.5%

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

   3. Simplified 9.5%

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

   5. Taylor expanded in alpha around inf 93.5%

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

      1. associate--l+ 93.5%

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

      2. sub-neg 93.5%

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

      3. +-commutative 93.5%

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

      4. mul-1-neg 93.5%

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

      5. unsub-neg 93.5%

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

      6. associate-/l* 93.5%

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

      7. distribute-rgt-neg-in 93.5%

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

      8. distribute-frac-neg 93.5%

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

      9. distribute-neg-in 93.5%

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

      10. metadata-eval 93.5%

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

      11. unsub-neg 93.5%

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

   7. Simplified 93.5%

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

   8. Taylor expanded in beta around 0 99.4%

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

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

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. div-inv 99.9%

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

      2. fma-define 100.0%

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

      3. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.96:\\ \;\;\;\;\frac{\frac{2 + \left(\left(\beta \cdot 2 + \left(4 \cdot \frac{-1}{\alpha} - \beta \cdot \left(4 \cdot \frac{1}{\alpha} + \frac{\beta}{\alpha}\right)\right)\right) + \beta \cdot \frac{-2 - \beta}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.96)
     (/
      (/
       (+
        2.0
        (+
         (+
          (* beta 2.0)
          (-
           (* 4.0 (/ -1.0 alpha))
           (* beta (+ (* 4.0 (/ 1.0 alpha)) (/ beta alpha)))))
         (* beta (/ (- -2.0 beta) alpha))))
       alpha)
      2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.96) {
		tmp = ((2.0 + (((beta * 2.0) + ((4.0 * (-1.0 / alpha)) - (beta * ((4.0 * (1.0 / alpha)) + (beta / alpha))))) + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.96d0)) then
        tmp = ((2.0d0 + (((beta * 2.0d0) + ((4.0d0 * ((-1.0d0) / alpha)) - (beta * ((4.0d0 * (1.0d0 / alpha)) + (beta / alpha))))) + (beta * (((-2.0d0) - beta) / alpha)))) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.96) {
		tmp = ((2.0 + (((beta * 2.0) + ((4.0 * (-1.0 / alpha)) - (beta * ((4.0 * (1.0 / alpha)) + (beta / alpha))))) + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.96:
		tmp = ((2.0 + (((beta * 2.0) + ((4.0 * (-1.0 / alpha)) - (beta * ((4.0 * (1.0 / alpha)) + (beta / alpha))))) + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.96)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(Float64(beta * 2.0) + Float64(Float64(4.0 * Float64(-1.0 / alpha)) - Float64(beta * Float64(Float64(4.0 * Float64(1.0 / alpha)) + Float64(beta / alpha))))) + Float64(beta * Float64(Float64(-2.0 - beta) / alpha)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.96)
		tmp = ((2.0 + (((beta * 2.0) + ((4.0 * (-1.0 / alpha)) - (beta * ((4.0 * (1.0 / alpha)) + (beta / alpha))))) + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.96], N[(N[(N[(2.0 + N[(N[(N[(beta * 2.0), $MachinePrecision] + N[(N[(4.0 * N[(-1.0 / alpha), $MachinePrecision]), $MachinePrecision] - N[(beta * N[(N[(4.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(beta * N[(N[(-2.0 - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 9.5%

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

      1. +-commutative 9.5%

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

   3. Simplified 9.5%

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

   5. Taylor expanded in alpha around inf 93.5%

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

      1. associate--l+ 93.5%

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

      2. sub-neg 93.5%

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

      3. +-commutative 93.5%

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

      4. mul-1-neg 93.5%

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

      5. unsub-neg 93.5%

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

      6. associate-/l* 93.5%

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

      7. distribute-rgt-neg-in 93.5%

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

      8. distribute-frac-neg 93.5%

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

      9. distribute-neg-in 93.5%

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

      10. metadata-eval 93.5%

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

      11. unsub-neg 93.5%

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

   7. Simplified 93.5%

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

   8. Taylor expanded in beta around 0 99.4%

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

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

   1. Initial program 100.0%

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

4. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.96:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot \left(\left(2 + \frac{\beta}{\alpha} \cdot -2\right) - \frac{6}{\alpha}\right)\right) + 4 \cdot \frac{-1}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.96)
     (/
      (/
       (+
        (+ 2.0 (* beta (- (+ 2.0 (* (/ beta alpha) -2.0)) (/ 6.0 alpha))))
        (* 4.0 (/ -1.0 alpha)))
       alpha)
      2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.96) {
		tmp = (((2.0 + (beta * ((2.0 + ((beta / alpha) * -2.0)) - (6.0 / alpha)))) + (4.0 * (-1.0 / alpha))) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.96d0)) then
        tmp = (((2.0d0 + (beta * ((2.0d0 + ((beta / alpha) * (-2.0d0))) - (6.0d0 / alpha)))) + (4.0d0 * ((-1.0d0) / alpha))) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.96) {
		tmp = (((2.0 + (beta * ((2.0 + ((beta / alpha) * -2.0)) - (6.0 / alpha)))) + (4.0 * (-1.0 / alpha))) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.96:
		tmp = (((2.0 + (beta * ((2.0 + ((beta / alpha) * -2.0)) - (6.0 / alpha)))) + (4.0 * (-1.0 / alpha))) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.96)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * Float64(Float64(2.0 + Float64(Float64(beta / alpha) * -2.0)) - Float64(6.0 / alpha)))) + Float64(4.0 * Float64(-1.0 / alpha))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.96)
		tmp = (((2.0 + (beta * ((2.0 + ((beta / alpha) * -2.0)) - (6.0 / alpha)))) + (4.0 * (-1.0 / alpha))) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.96], N[(N[(N[(N[(2.0 + N[(beta * N[(N[(2.0 + N[(N[(beta / alpha), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] - N[(6.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(-1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 9.5%

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

      1. +-commutative 9.5%

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

   3. Simplified 9.5%

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

   5. Taylor expanded in alpha around inf 93.5%

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

      1. associate--l+ 93.5%

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

      2. sub-neg 93.5%

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

      3. +-commutative 93.5%

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

      4. mul-1-neg 93.5%

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

      5. unsub-neg 93.5%

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

      6. associate-/l* 93.5%

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

      7. distribute-rgt-neg-in 93.5%

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

      8. distribute-frac-neg 93.5%

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

      9. distribute-neg-in 93.5%

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

      10. metadata-eval 93.5%

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

      11. unsub-neg 93.5%

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

   7. Simplified 93.5%

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

   8. Taylor expanded in beta around 0 99.4%

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

   9. Taylor expanded in alpha around 0 99.4%

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

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

   1. Initial program 100.0%

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

4. Final simplification 99.8%

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

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.96:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot \frac{-2 - \beta}{\alpha} + \left(\beta \cdot 2 - \frac{4}{\alpha}\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.96)
     (/
      (/
       (+
        2.0
        (+ (* beta (/ (- -2.0 beta) alpha)) (- (* beta 2.0) (/ 4.0 alpha))))
       alpha)
      2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.96) {
		tmp = ((2.0 + ((beta * ((-2.0 - beta) / alpha)) + ((beta * 2.0) - (4.0 / alpha)))) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.96d0)) then
        tmp = ((2.0d0 + ((beta * (((-2.0d0) - beta) / alpha)) + ((beta * 2.0d0) - (4.0d0 / alpha)))) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.96) {
		tmp = ((2.0 + ((beta * ((-2.0 - beta) / alpha)) + ((beta * 2.0) - (4.0 / alpha)))) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.96:
		tmp = ((2.0 + ((beta * ((-2.0 - beta) / alpha)) + ((beta * 2.0) - (4.0 / alpha)))) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.96)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * Float64(Float64(-2.0 - beta) / alpha)) + Float64(Float64(beta * 2.0) - Float64(4.0 / alpha)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.96)
		tmp = ((2.0 + ((beta * ((-2.0 - beta) / alpha)) + ((beta * 2.0) - (4.0 / alpha)))) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.96], N[(N[(N[(2.0 + N[(N[(beta * N[(N[(-2.0 - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(beta * 2.0), $MachinePrecision] - N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 9.5%

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

      1. +-commutative 9.5%

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

   3. Simplified 9.5%

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

   5. Taylor expanded in alpha around inf 93.5%

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

      1. associate--l+ 93.5%

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

      2. sub-neg 93.5%

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

      3. +-commutative 93.5%

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

      4. mul-1-neg 93.5%

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

      5. unsub-neg 93.5%

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

      6. associate-/l* 93.5%

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

      7. distribute-rgt-neg-in 93.5%

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

      8. distribute-frac-neg 93.5%

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

      9. distribute-neg-in 93.5%

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

      10. metadata-eval 93.5%

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

      11. unsub-neg 93.5%

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

   7. Simplified 93.5%

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

   8. Taylor expanded in beta around 0 98.9%

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

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

   1. Initial program 100.0%

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

4. Final simplification 99.7%

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

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99999999:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999999) (/ (+ beta 1.0) alpha) (/ (+ 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 <= -0.99999999) {
		tmp = (beta + 1.0) / alpha;
	} 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 <= (-0.99999999d0)) then
        tmp = (beta + 1.0d0) / alpha
    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 <= -0.99999999) {
		tmp = (beta + 1.0) / alpha;
	} 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 <= -0.99999999:
		tmp = (beta + 1.0) / alpha
	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 <= -0.99999999)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	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 <= -0.99999999)
		tmp = (beta + 1.0) / alpha;
	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, -0.99999999], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99999998999999995

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

   3. Simplified 6.7%

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

   5. Taylor expanded in alpha around inf 93.8%

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

      1. associate--l+ 93.8%

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

      2. sub-neg 93.8%

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

      3. +-commutative 93.8%

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

      4. mul-1-neg 93.8%

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

      5. unsub-neg 93.8%

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

      6. associate-/l* 93.8%

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

      7. distribute-rgt-neg-in 93.8%

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

      8. distribute-frac-neg 93.8%

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

      9. distribute-neg-in 93.8%

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

      10. metadata-eval 93.8%

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

      11. unsub-neg 93.8%

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

   7. Simplified 93.8%

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

      1. clear-num 93.8%

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

      2. inv-pow 93.8%

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

      3. associate-+l- 93.8%

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

      4. *-commutative 93.8%

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

      5. associate-*r/ 93.8%

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

      6. sub-div 93.8%

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

      7. +-commutative 93.8%

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

   9. Applied egg-rr 93.8%

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

       1. unpow-1 93.8%

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

       2. associate-/r/ 93.8%

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

       3. associate-+r- 93.8%

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

       4. *-commutative 93.8%

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

       5. +-commutative 93.8%

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

       6. fma-undefine 93.8%

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

       7. +-commutative 93.8%

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

   11. Simplified 93.8%

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

   12. Taylor expanded in alpha around -inf 93.7%

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

   13. Taylor expanded in alpha around inf 98.9%

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

       1. associate-*r/ 98.9%

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

       2. distribute-lft-in 98.9%

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

       3. metadata-eval 98.9%

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

       4. associate-*r* 98.9%

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

       5. metadata-eval 98.9%

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

       6. *-lft-identity 98.9%

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

   15. Simplified 98.9%

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

   if -0.99999998999999995 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999999:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \beta \cdot 0.25\\ \mathbf{if}\;\beta \leq -6.4 \cdot 10^{-242}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* beta 0.25))))
   (if (<= beta -6.4e-242)
     t_0
     (if (<= beta 1.6e-278)
       (/ 1.0 alpha)
       (if (<= beta 2.0) t_0 (+ 1.0 (/ -1.0 beta)))))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (beta * 0.25);
	double tmp;
	if (beta <= -6.4e-242) {
		tmp = t_0;
	} else if (beta <= 1.6e-278) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (beta * 0.25d0)
    if (beta <= (-6.4d-242)) then
        tmp = t_0
    else if (beta <= 1.6d-278) then
        tmp = 1.0d0 / alpha
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (beta * 0.25);
	double tmp;
	if (beta <= -6.4e-242) {
		tmp = t_0;
	} else if (beta <= 1.6e-278) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (beta * 0.25)
	tmp = 0
	if beta <= -6.4e-242:
		tmp = t_0
	elif beta <= 1.6e-278:
		tmp = 1.0 / alpha
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(beta * 0.25))
	tmp = 0.0
	if (beta <= -6.4e-242)
		tmp = t_0;
	elseif (beta <= 1.6e-278)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (beta * 0.25);
	tmp = 0.0;
	if (beta <= -6.4e-242)
		tmp = t_0;
	elseif (beta <= 1.6e-278)
		tmp = 1.0 / alpha;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0 + (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, -6.4e-242], t$95$0, If[LessEqual[beta, 1.6e-278], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 2.0], t$95$0, N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < -6.39999999999999997e-242 or 1.60000000000000009e-278 < beta < 2

   1. Initial program 77.0%

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

      1. +-commutative 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in alpha around 0 73.6%

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

   6. Taylor expanded in beta around 0 71.8%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \beta} \]
   7. Step-by-step derivation

      1. *-commutative 71.8%

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

   8. Simplified 71.8%

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

   if -6.39999999999999997e-242 < beta < 1.60000000000000009e-278

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

   3. Simplified 41.6%

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

   5. Taylor expanded in beta around 0 41.6%

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

      1. +-commutative 41.6%

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

   7. Simplified 41.6%

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

   8. Taylor expanded in alpha around inf 63.8%

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

   if 2 < beta

   1. Initial program 83.7%

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

      1. +-commutative 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in alpha around 0 83.4%

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

   6. Taylor expanded in beta around inf 83.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -6.4 \cdot 10^{-242}:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \beta \cdot 0.25\\ \mathbf{if}\;\beta \leq -6.4 \cdot 10^{-242}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* beta 0.25))))
   (if (<= beta -6.4e-242)
     t_0
     (if (<= beta 1.6e-278) (/ 1.0 alpha) (if (<= beta 2.0) t_0 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (beta * 0.25);
	double tmp;
	if (beta <= -6.4e-242) {
		tmp = t_0;
	} else if (beta <= 1.6e-278) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (beta * 0.25d0)
    if (beta <= (-6.4d-242)) then
        tmp = t_0
    else if (beta <= 1.6d-278) then
        tmp = 1.0d0 / alpha
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (beta * 0.25);
	double tmp;
	if (beta <= -6.4e-242) {
		tmp = t_0;
	} else if (beta <= 1.6e-278) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (beta * 0.25)
	tmp = 0
	if beta <= -6.4e-242:
		tmp = t_0
	elif beta <= 1.6e-278:
		tmp = 1.0 / alpha
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(beta * 0.25))
	tmp = 0.0
	if (beta <= -6.4e-242)
		tmp = t_0;
	elseif (beta <= 1.6e-278)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (beta * 0.25);
	tmp = 0.0;
	if (beta <= -6.4e-242)
		tmp = t_0;
	elseif (beta <= 1.6e-278)
		tmp = 1.0 / alpha;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, -6.4e-242], t$95$0, If[LessEqual[beta, 1.6e-278], 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 < -6.39999999999999997e-242 or 1.60000000000000009e-278 < beta < 2

   1. Initial program 77.0%

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

      1. +-commutative 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in alpha around 0 73.6%

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

   6. Taylor expanded in beta around 0 71.8%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \beta} \]
   7. Step-by-step derivation

      1. *-commutative 71.8%

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

   8. Simplified 71.8%

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

   if -6.39999999999999997e-242 < beta < 1.60000000000000009e-278

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

   3. Simplified 41.6%

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

   5. Taylor expanded in beta around 0 41.6%

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

      1. +-commutative 41.6%

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

   7. Simplified 41.6%

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

   8. Taylor expanded in alpha around inf 63.8%

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

   if 2 < beta

   1. Initial program 83.7%

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

      1. +-commutative 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in beta around inf 82.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -6.4 \cdot 10^{-242}:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq -6.4 \cdot 10^{-242}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta -6.4e-242)
   (+ 0.5 (* alpha -0.25))
   (if (<= beta 1.6e-278) (/ 1.0 alpha) (if (<= beta 2.0) 0.5 1.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= -6.4e-242) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (beta <= 1.6e-278) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= (-6.4d-242)) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else if (beta <= 1.6d-278) then
        tmp = 1.0d0 / alpha
    else if (beta <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= -6.4e-242) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (beta <= 1.6e-278) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= -6.4e-242:
		tmp = 0.5 + (alpha * -0.25)
	elif beta <= 1.6e-278:
		tmp = 1.0 / alpha
	elif beta <= 2.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= -6.4e-242)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	elseif (beta <= 1.6e-278)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= -6.4e-242)
		tmp = 0.5 + (alpha * -0.25);
	elseif (beta <= 1.6e-278)
		tmp = 1.0 / alpha;
	elseif (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, -6.4e-242], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.6e-278], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 2.0], 0.5, 1.0]]]

Derivation

1. Split input into 4 regimes

2. if beta < -6.39999999999999997e-242

   1. Initial program 81.8%

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

      1. +-commutative 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in beta around 0 78.9%

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

      1. +-commutative 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in alpha around 0 76.0%

      \[\leadsto \color{blue}{0.5 + -0.25 \cdot \alpha} \]
   9. Step-by-step derivation

      1. *-commutative 76.0%

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

   10. Simplified 76.0%

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

   if -6.39999999999999997e-242 < beta < 1.60000000000000009e-278

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

   3. Simplified 41.6%

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

   5. Taylor expanded in beta around 0 41.6%

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

      1. +-commutative 41.6%

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

   7. Simplified 41.6%

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

   8. Taylor expanded in alpha around inf 63.8%

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

   if 1.60000000000000009e-278 < beta < 2

   1. Initial program 73.8%

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

      1. +-commutative 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in alpha around 0 70.8%

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

   6. Taylor expanded in beta around 0 67.6%

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

   if 2 < beta

   1. Initial program 83.7%

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

      1. +-commutative 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in beta around inf 82.8%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -6.4 \cdot 10^{-242}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq -6.4 \cdot 10^{-242}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta -6.4e-242)
   0.5
   (if (<= beta 1.6e-278) (/ 1.0 alpha) (if (<= beta 2.0) 0.5 1.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= -6.4e-242) {
		tmp = 0.5;
	} else if (beta <= 1.6e-278) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= (-6.4d-242)) then
        tmp = 0.5d0
    else if (beta <= 1.6d-278) then
        tmp = 1.0d0 / alpha
    else if (beta <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= -6.4e-242) {
		tmp = 0.5;
	} else if (beta <= 1.6e-278) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= -6.4e-242:
		tmp = 0.5
	elif beta <= 1.6e-278:
		tmp = 1.0 / alpha
	elif beta <= 2.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= -6.4e-242)
		tmp = 0.5;
	elseif (beta <= 1.6e-278)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= -6.4e-242)
		tmp = 0.5;
	elseif (beta <= 1.6e-278)
		tmp = 1.0 / alpha;
	elseif (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, -6.4e-242], 0.5, If[LessEqual[beta, 1.6e-278], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 2.0], 0.5, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if beta < -6.39999999999999997e-242 or 1.60000000000000009e-278 < beta < 2

   1. Initial program 77.0%

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

      1. +-commutative 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in alpha around 0 73.6%

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

   6. Taylor expanded in beta around 0 70.5%

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

   if -6.39999999999999997e-242 < beta < 1.60000000000000009e-278

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

   3. Simplified 41.6%

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

   5. Taylor expanded in beta around 0 41.6%

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

      1. +-commutative 41.6%

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

   7. Simplified 41.6%

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

   8. Taylor expanded in alpha around inf 63.8%

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

   if 2 < beta

   1. Initial program 83.7%

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

      1. +-commutative 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in beta around inf 82.8%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -6.4 \cdot 10^{-242}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 1

   1. Initial program 72.6%

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

      1. +-commutative 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in alpha around inf 30.9%

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

      1. associate--l+ 30.9%

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

      2. sub-neg 30.9%

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

      3. +-commutative 30.9%

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

      4. mul-1-neg 30.9%

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

      5. unsub-neg 30.9%

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

      6. associate-/l* 30.9%

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

      7. distribute-rgt-neg-in 30.9%

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

      8. distribute-frac-neg 30.9%

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

      9. distribute-neg-in 30.9%

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

      10. metadata-eval 30.9%

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

      11. unsub-neg 30.9%

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

   7. Simplified 30.9%

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

      1. clear-num 30.9%

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

      2. inv-pow 30.9%

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

      3. associate-+l- 30.9%

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

      4. *-commutative 30.9%

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

      5. associate-*r/ 30.9%

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

      6. sub-div 30.9%

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

      7. +-commutative 30.9%

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

   9. Applied egg-rr 30.9%

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

       1. unpow-1 30.9%

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

       2. associate-/r/ 30.9%

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

       3. associate-+r- 30.9%

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

       4. *-commutative 30.9%

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

       5. +-commutative 30.9%

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

       6. fma-undefine 30.9%

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

       7. +-commutative 30.9%

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

   11. Simplified 30.9%

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

   12. Taylor expanded in alpha around -inf 99.7%

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

   13. Taylor expanded in beta around 0 96.3%

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

       1. distribute-lft-in 96.3%

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

       2. *-rgt-identity 96.3%

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

       3. *-commutative 96.3%

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

       4. associate-*r* 96.3%

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

       5. rgt-mult-inverse 96.5%

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

       6. metadata-eval 96.5%

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

   15. Simplified 96.5%

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

   if 1 < beta

   1. Initial program 83.7%

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

      1. +-commutative 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in alpha around inf 13.5%

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

      1. associate--l+ 13.5%

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

      2. sub-neg 13.5%

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

      3. +-commutative 13.5%

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

      4. mul-1-neg 13.5%

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

      5. unsub-neg 13.5%

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

      6. associate-/l* 13.5%

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

      7. distribute-rgt-neg-in 13.5%

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

      8. distribute-frac-neg 13.5%

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

      9. distribute-neg-in 13.5%

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

      10. metadata-eval 13.5%

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

      11. unsub-neg 13.5%

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

   7. Simplified 13.5%

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

      1. clear-num 13.5%

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

      2. inv-pow 13.5%

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

      3. associate-+l- 13.5%

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

      4. *-commutative 13.5%

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

      5. associate-*r/ 13.5%

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

      6. sub-div 13.5%

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

      7. +-commutative 13.5%

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

   9. Applied egg-rr 13.5%

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

       1. unpow-1 13.5%

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

       2. associate-/r/ 13.5%

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

       3. associate-+r- 13.5%

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

       4. *-commutative 13.5%

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

       5. +-commutative 13.5%

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

       6. fma-undefine 13.5%

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

       7. +-commutative 13.5%

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

   11. Simplified 13.5%

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

   12. Taylor expanded in alpha around -inf 50.4%

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

   13. Taylor expanded in beta around inf 99.8%

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

       1. +-commutative 99.8%

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

       2. distribute-lft-in 99.8%

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

       3. rgt-mult-inverse 99.8%

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

       4. *-rgt-identity 99.8%

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

   15. Simplified 99.8%

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

4. Final simplification 97.6%

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

Alternative 11: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 1.0d0 / (alpha + 2.0d0)
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 72.6%

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

      1. +-commutative 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in alpha around inf 30.9%

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

      1. associate--l+ 30.9%

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

      2. sub-neg 30.9%

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

      3. +-commutative 30.9%

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

      4. mul-1-neg 30.9%

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

      5. unsub-neg 30.9%

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

      6. associate-/l* 30.9%

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

      7. distribute-rgt-neg-in 30.9%

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

      8. distribute-frac-neg 30.9%

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

      9. distribute-neg-in 30.9%

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

      10. metadata-eval 30.9%

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

      11. unsub-neg 30.9%

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

   7. Simplified 30.9%

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

      1. clear-num 30.9%

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

      2. inv-pow 30.9%

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

      3. associate-+l- 30.9%

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

      4. *-commutative 30.9%

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

      5. associate-*r/ 30.9%

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

      6. sub-div 30.9%

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

      7. +-commutative 30.9%

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

   9. Applied egg-rr 30.9%

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

       1. unpow-1 30.9%

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

       2. associate-/r/ 30.9%

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

       3. associate-+r- 30.9%

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

       4. *-commutative 30.9%

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

       5. +-commutative 30.9%

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

       6. fma-undefine 30.9%

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

       7. +-commutative 30.9%

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

   11. Simplified 30.9%

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

   12. Taylor expanded in alpha around -inf 99.7%

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

   13. Taylor expanded in beta around 0 96.3%

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

       1. distribute-lft-in 96.3%

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

       2. *-rgt-identity 96.3%

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

       3. *-commutative 96.3%

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

       4. associate-*r* 96.3%

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

       5. rgt-mult-inverse 96.5%

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

       6. metadata-eval 96.5%

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

   15. Simplified 96.5%

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

   if 6 < beta

   1. Initial program 83.7%

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

      1. +-commutative 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in alpha around 0 83.4%

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

   6. Taylor expanded in beta around inf 83.5%

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

4. Final simplification 92.2%

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

Alternative 12: 69.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.55:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.55) 0.5 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.55) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.55d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.55) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.55:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.55)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.55)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.55], 0.5, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.55000000000000004

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.3%

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

   6. Taylor expanded in beta around 0 69.5%

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

   if 1.55000000000000004 < alpha

   1. Initial program 26.0%

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

      1. +-commutative 26.0%

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

   3. Simplified 26.0%

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

   5. Taylor expanded in beta around 0 8.8%

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

      1. +-commutative 8.8%

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

   7. Simplified 8.8%

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

   8. Taylor expanded in alpha around inf 61.9%

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

Alternative 13: 49.4% 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 76.3%

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

   1. +-commutative 76.3%

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

3. Simplified 76.3%

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

5. Taylor expanded in alpha around 0 73.8%

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

6. Taylor expanded in beta around 0 49.8%

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

Reproduce

herbie shell --seed 2024116 
(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))