Octave 3.8, jcobi/1

Percentage Accurate: 74.7% → 99.9%

Time: 9.7s

Alternatives: 11

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{2 - \left(\beta \cdot \frac{\beta - -2}{\alpha} - \left(\beta \cdot 2 - \left(\beta + 2\right) \cdot \frac{\beta + 2}{\alpha}\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99995)
   (/
    (/
     (-
      2.0
      (-
       (* beta (/ (- beta -2.0) alpha))
       (- (* beta 2.0) (* (+ beta 2.0) (/ (+ beta 2.0) 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)) <= -0.99995) {
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) - ((beta * 2.0) - ((beta + 2.0) * ((beta + 2.0) / 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)) <= -0.99995)
		tmp = Float64(Float64(Float64(2.0 - Float64(Float64(beta * Float64(Float64(beta - -2.0) / alpha)) - Float64(Float64(beta * 2.0) - Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) / 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], -0.99995], N[(N[(N[(2.0 - N[(N[(beta * N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] - N[(N[(beta * 2.0), $MachinePrecision] - N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $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))) < -0.999950000000000006

   1. Initial program 8.3%

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

      1. +-commutative 8.3%

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

   3. Simplified 8.3%

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

   5. Taylor expanded in alpha around inf 97.2%

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

      1. associate--l+ 97.2%

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

      2. sub-neg 97.2%

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

      3. +-commutative 97.2%

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

      4. mul-1-neg 97.2%

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

      5. unsub-neg 97.2%

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

      6. associate-/l* 97.2%

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

      7. distribute-rgt-neg-in 97.2%

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

      8. distribute-frac-neg 97.2%

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

      9. distribute-neg-in 97.2%

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

      10. metadata-eval 97.2%

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

      11. unsub-neg 97.2%

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

   7. Simplified 97.2%

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

      1. unpow2 97.2%

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

      2. *-un-lft-identity 97.2%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

   if -0.999950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]
   4. 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 99.9%

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

      3. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{2 - \left(\beta \cdot \frac{\beta - -2}{\alpha} - \left(\beta \cdot 2 - \left(\beta + 2\right) \cdot \frac{\beta + 2}{\alpha}\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{2 - \left(\beta \cdot \frac{\beta - -2}{\alpha} - \left(\beta \cdot 2 - \left(\beta + 2\right) \cdot \frac{\beta + 2}{\alpha}\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99995)
   (/
    (/
     (-
      2.0
      (-
       (* beta (/ (- beta -2.0) alpha))
       (- (* beta 2.0) (* (+ beta 2.0) (/ (+ beta 2.0) alpha)))))
     alpha)
    2.0)
   (/ (+ 1.0 (/ (- beta alpha) (+ alpha (+ beta 2.0)))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) - ((beta * 2.0) - ((beta + 2.0) * ((beta + 2.0) / alpha))))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99995d0)) then
        tmp = ((2.0d0 - ((beta * ((beta - (-2.0d0)) / alpha)) - ((beta * 2.0d0) - ((beta + 2.0d0) * ((beta + 2.0d0) / alpha))))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / (alpha + (beta + 2.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) - ((beta * 2.0) - ((beta + 2.0) * ((beta + 2.0) / alpha))))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995:
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) - ((beta * 2.0) - ((beta + 2.0) * ((beta + 2.0) / alpha))))) / alpha) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99995)
		tmp = Float64(Float64(Float64(2.0 - Float64(Float64(beta * Float64(Float64(beta - -2.0) / alpha)) - Float64(Float64(beta * 2.0) - Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) / alpha))))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995)
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) - ((beta * 2.0) - ((beta + 2.0) * ((beta + 2.0) / alpha))))) / alpha) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99995], N[(N[(N[(2.0 - N[(N[(beta * N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] - N[(N[(beta * 2.0), $MachinePrecision] - N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -0.999950000000000006

   1. Initial program 8.3%

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

      1. +-commutative 8.3%

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

   3. Simplified 8.3%

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

   5. Taylor expanded in alpha around inf 97.2%

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

      1. associate--l+ 97.2%

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

      2. sub-neg 97.2%

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

      3. +-commutative 97.2%

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

      4. mul-1-neg 97.2%

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

      5. unsub-neg 97.2%

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

      6. associate-/l* 97.2%

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

      7. distribute-rgt-neg-in 97.2%

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

      8. distribute-frac-neg 97.2%

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

      9. distribute-neg-in 97.2%

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

      10. metadata-eval 97.2%

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

      11. unsub-neg 97.2%

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

   7. Simplified 97.2%

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

      1. unpow2 97.2%

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

      2. *-un-lft-identity 97.2%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

   if -0.999950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.9%

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

      1. associate-+r+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{2 - \left(\beta \cdot \frac{\beta - -2}{\alpha} - \left(\beta \cdot 2 - \left(\beta + 2\right) \cdot \frac{\beta + 2}{\alpha}\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{1}{\alpha \cdot \left(\frac{0.5}{\alpha} - \frac{-1}{2 + \beta \cdot 2}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99995)
   (/ (/ 1.0 (* alpha (- (/ 0.5 alpha) (/ -1.0 (+ 2.0 (* beta 2.0)))))) 2.0)
   (/ (+ 1.0 (/ (- beta alpha) (+ alpha (+ beta 2.0)))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = (1.0 / (alpha * ((0.5 / alpha) - (-1.0 / (2.0 + (beta * 2.0)))))) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99995d0)) then
        tmp = (1.0d0 / (alpha * ((0.5d0 / alpha) - ((-1.0d0) / (2.0d0 + (beta * 2.0d0)))))) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / (alpha + (beta + 2.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = (1.0 / (alpha * ((0.5 / alpha) - (-1.0 / (2.0 + (beta * 2.0)))))) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995:
		tmp = (1.0 / (alpha * ((0.5 / alpha) - (-1.0 / (2.0 + (beta * 2.0)))))) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99995)
		tmp = Float64(Float64(1.0 / Float64(alpha * Float64(Float64(0.5 / alpha) - Float64(-1.0 / Float64(2.0 + Float64(beta * 2.0)))))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995)
		tmp = (1.0 / (alpha * ((0.5 / alpha) - (-1.0 / (2.0 + (beta * 2.0)))))) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99995], N[(N[(1.0 / N[(alpha * N[(N[(0.5 / alpha), $MachinePrecision] - N[(-1.0 / N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -0.999950000000000006

   1. Initial program 8.3%

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

      1. +-commutative 8.3%

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

   3. Simplified 8.3%

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

   5. Taylor expanded in alpha around inf 97.2%

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

      1. associate--l+ 97.2%

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

      2. sub-neg 97.2%

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

      3. +-commutative 97.2%

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

      4. mul-1-neg 97.2%

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

      5. unsub-neg 97.2%

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

      6. associate-/l* 97.2%

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

      7. distribute-rgt-neg-in 97.2%

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

      8. distribute-frac-neg 97.2%

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

      9. distribute-neg-in 97.2%

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

      10. metadata-eval 97.2%

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

      11. unsub-neg 97.2%

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

   7. Simplified 97.2%

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

      1. clear-num 97.2%

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

      2. inv-pow 97.2%

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

      3. associate-+l- 97.2%

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

      4. *-commutative 97.2%

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

      5. associate-*r/ 97.2%

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

      6. sub-div 97.2%

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

      7. +-commutative 97.2%

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

   9. Applied egg-rr 97.2%

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

       1. unpow-1 97.2%

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

       2. *-commutative 97.2%

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

       3. +-commutative 97.2%

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

   11. Simplified 97.2%

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

   12. Taylor expanded in alpha around inf 97.1%

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

       1. associate--l+ 97.1%

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

       2. associate-*r/ 97.1%

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

       3. div-sub 97.1%

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

       4. associate-/r* 97.1%

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

       5. associate-*r* 97.1%

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

       6. mul-1-neg 97.1%

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

   14. Simplified 97.1%

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

   15. Taylor expanded in beta around inf 98.6%

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

   if -0.999950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in beta around 0 99.9%

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

      1. associate-+r+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{1}{\alpha \cdot \left(\frac{0.5}{\alpha} - \frac{-1}{2 + \beta \cdot 2}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999999)
   (/ (/ (+ beta (- beta -2.0)) alpha) 2.0)
   (/ (+ 1.0 (/ (- beta alpha) (+ alpha (+ beta 2.0)))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999999) {
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99999999d0)) then
        tmp = ((beta + (beta - (-2.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / (alpha + (beta + 2.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999999) {
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999999:
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999999)
		tmp = Float64(Float64(Float64(beta + Float64(beta - -2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999999)
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999999], N[(N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 7.6%

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

      1. +-commutative 7.6%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in alpha around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-neg-frac2 98.7%

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

      3. sub-neg 98.7%

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

      4. +-commutative 98.7%

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

      5. mul-1-neg 98.7%

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

      6. sub-neg 98.7%

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

      7. mul-1-neg 98.7%

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

      8. distribute-lft-in 98.7%

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

      9. metadata-eval 98.7%

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

      10. mul-1-neg 98.7%

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

      11. unsub-neg 98.7%

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

   7. Simplified 98.7%

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

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

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in beta around 0 99.7%

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

      1. associate-+r+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.4%

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

Alternative 5: 60.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq -1.8 \cdot 10^{-126}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 3.35 \cdot 10^{-141}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\beta}}{2}\\ \mathbf{elif}\;\alpha \leq 33000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha -1.8e-126)
   1.0
   (if (<= alpha 3.35e-141)
     (/ (- 1.0 (/ alpha beta)) 2.0)
     (if (<= alpha 33000.0) 1.0 (/ (+ beta 1.0) alpha)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -1.8e-126) {
		tmp = 1.0;
	} else if (alpha <= 3.35e-141) {
		tmp = (1.0 - (alpha / beta)) / 2.0;
	} else if (alpha <= 33000.0) {
		tmp = 1.0;
	} else {
		tmp = (beta + 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 <= (-1.8d-126)) then
        tmp = 1.0d0
    else if (alpha <= 3.35d-141) then
        tmp = (1.0d0 - (alpha / beta)) / 2.0d0
    else if (alpha <= 33000.0d0) then
        tmp = 1.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -1.8e-126) {
		tmp = 1.0;
	} else if (alpha <= 3.35e-141) {
		tmp = (1.0 - (alpha / beta)) / 2.0;
	} else if (alpha <= 33000.0) {
		tmp = 1.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= -1.8e-126:
		tmp = 1.0
	elif alpha <= 3.35e-141:
		tmp = (1.0 - (alpha / beta)) / 2.0
	elif alpha <= 33000.0:
		tmp = 1.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= -1.8e-126)
		tmp = 1.0;
	elseif (alpha <= 3.35e-141)
		tmp = Float64(Float64(1.0 - Float64(alpha / beta)) / 2.0);
	elseif (alpha <= 33000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= -1.8e-126)
		tmp = 1.0;
	elseif (alpha <= 3.35e-141)
		tmp = (1.0 - (alpha / beta)) / 2.0;
	elseif (alpha <= 33000.0)
		tmp = 1.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, -1.8e-126], 1.0, If[LessEqual[alpha, 3.35e-141], N[(N[(1.0 - N[(alpha / beta), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 33000.0], 1.0, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -1.8e-126 or 3.3500000000000002e-141 < alpha < 33000

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 54.0%

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

   if -1.8e-126 < alpha < 3.3500000000000002e-141

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 39.3%

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

   6. Taylor expanded in beta around 0 57.2%

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

      1. associate-*r/ 57.2%

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

      2. mul-1-neg 57.2%

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

   8. Simplified 57.2%

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

   if 33000 < alpha

   1. Initial program 26.2%

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

      1. +-commutative 26.2%

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

   3. Simplified 26.2%

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

   5. Taylor expanded in alpha around inf 78.8%

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

      1. associate--l+ 78.8%

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

      2. sub-neg 78.8%

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

      3. +-commutative 78.8%

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

      4. mul-1-neg 78.8%

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

      5. unsub-neg 78.8%

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

      6. associate-/l* 78.8%

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

      7. distribute-rgt-neg-in 78.8%

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

      8. distribute-frac-neg 78.8%

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

      9. distribute-neg-in 78.8%

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

      10. metadata-eval 78.8%

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

      11. unsub-neg 78.8%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in alpha around inf 80.4%

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

      1. associate-*r/ 80.4%

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

      2. distribute-lft-in 80.4%

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

      3. metadata-eval 80.4%

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

      4. associate-*r* 80.4%

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

      5. metadata-eval 80.4%

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

      6. *-lft-identity 80.4%

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

   10. Simplified 80.4%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -1.8 \cdot 10^{-126}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 3.35 \cdot 10^{-141}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\beta}}{2}\\ \mathbf{elif}\;\alpha \leq 33000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{1}{\alpha \cdot \left(0.5 + \frac{1}{\alpha}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.2e-19)
   (/ (/ 1.0 (* alpha (+ 0.5 (/ 1.0 alpha)))) 2.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2e-19) {
		tmp = (1.0 / (alpha * (0.5 + (1.0 / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.2d-19) then
        tmp = (1.0d0 / (alpha * (0.5d0 + (1.0d0 / alpha)))) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.2e-19) {
		tmp = (1.0 / (alpha * (0.5 + (1.0 / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 8.2e-19:
		tmp = (1.0 / (alpha * (0.5 + (1.0 / alpha)))) / 2.0
	else:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.2e-19)
		tmp = Float64(Float64(1.0 / Float64(alpha * Float64(0.5 + Float64(1.0 / alpha)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.2e-19)
		tmp = (1.0 / (alpha * (0.5 + (1.0 / alpha)))) / 2.0;
	else
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.2e-19], N[(N[(1.0 / N[(alpha * N[(0.5 + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.1999999999999997e-19

   1. Initial program 65.9%

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

      1. +-commutative 65.9%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in alpha around inf 37.8%

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

      1. associate--l+ 37.8%

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

      2. sub-neg 37.8%

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

      3. +-commutative 37.8%

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

      4. mul-1-neg 37.8%

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

      5. unsub-neg 37.8%

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

      6. associate-/l* 37.8%

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

      7. distribute-rgt-neg-in 37.8%

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

      8. distribute-frac-neg 37.8%

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

      9. distribute-neg-in 37.8%

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

      10. metadata-eval 37.8%

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

      11. unsub-neg 37.8%

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

   7. Simplified 37.8%

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

      1. clear-num 37.8%

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

      2. inv-pow 37.8%

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

      3. associate-+l- 37.8%

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

      4. *-commutative 37.8%

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

      5. associate-*r/ 37.8%

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

      6. sub-div 37.8%

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

      7. +-commutative 37.8%

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

   9. Applied egg-rr 37.8%

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

       1. unpow-1 37.8%

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

       2. *-commutative 37.8%

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

       3. +-commutative 37.8%

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

   11. Simplified 37.8%

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

   12. Taylor expanded in alpha around inf 99.8%

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

       1. associate--l+ 99.8%

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

       2. associate-*r/ 99.8%

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

       3. div-sub 99.8%

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

       4. associate-/r* 99.8%

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

       5. associate-*r* 99.8%

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

       6. mul-1-neg 99.8%

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

   14. Simplified 99.8%

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

   15. Taylor expanded in beta around 0 99.7%

       \[\leadsto \frac{\frac{1}{\color{blue}{\alpha \cdot \left(0.5 + \frac{1}{\alpha}\right)}}}{2} \]
   16. Step-by-step derivation

       1. +-commutative 99.7%

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

   17. Simplified 99.7%

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

   if 8.1999999999999997e-19 < beta

   1. Initial program 88.7%

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

      1. +-commutative 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in alpha around 0 88.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{1}{\alpha \cdot \left(0.5 + \frac{1}{\alpha}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 29000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 29000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta (- beta -2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 29000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 29000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + (beta - (-2.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 29000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 29000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 29000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta - -2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 29000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 29000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 29000

   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 98.8%

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

   if 29000 < alpha

   1. Initial program 26.2%

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

      1. +-commutative 26.2%

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

   3. Simplified 26.2%

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

   5. Taylor expanded in alpha around -inf 80.4%

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

      1. mul-1-neg 80.4%

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

      2. distribute-neg-frac2 80.4%

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

      3. sub-neg 80.4%

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

      4. +-commutative 80.4%

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

      5. mul-1-neg 80.4%

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

      6. sub-neg 80.4%

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

      7. mul-1-neg 80.4%

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

      8. distribute-lft-in 80.4%

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

      9. metadata-eval 80.4%

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

      10. mul-1-neg 80.4%

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

      11. unsub-neg 80.4%

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

   7. Simplified 80.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 29000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 25000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 25000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 25000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 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 <= 25000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 25000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 25000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 25000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 25000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 25000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 25000

   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 98.8%

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

   if 25000 < alpha

   1. Initial program 26.2%

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

      1. +-commutative 26.2%

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

   3. Simplified 26.2%

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

   5. Taylor expanded in alpha around inf 78.8%

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

      1. associate--l+ 78.8%

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

      2. sub-neg 78.8%

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

      3. +-commutative 78.8%

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

      4. mul-1-neg 78.8%

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

      5. unsub-neg 78.8%

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

      6. associate-/l* 78.8%

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

      7. distribute-rgt-neg-in 78.8%

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

      8. distribute-frac-neg 78.8%

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

      9. distribute-neg-in 78.8%

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

      10. metadata-eval 78.8%

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

      11. unsub-neg 78.8%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in alpha around inf 80.4%

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

      1. associate-*r/ 80.4%

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

      2. distribute-lft-in 80.4%

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

      3. metadata-eval 80.4%

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

      4. associate-*r* 80.4%

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

      5. metadata-eval 80.4%

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

      6. *-lft-identity 80.4%

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

   10. Simplified 80.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 25000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 32000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 32000.0) 1.0 (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 32000.0) {
		tmp = 1.0;
	} else {
		tmp = (beta + 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 <= 32000.0d0) then
        tmp = 1.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 32000.0) {
		tmp = 1.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 32000.0:
		tmp = 1.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 32000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 32000.0)
		tmp = 1.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 32000.0], 1.0, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 32000

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 47.0%

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

   if 32000 < alpha

   1. Initial program 26.2%

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

      1. +-commutative 26.2%

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

   3. Simplified 26.2%

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

   5. Taylor expanded in alpha around inf 78.8%

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

      1. associate--l+ 78.8%

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

      2. sub-neg 78.8%

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

      3. +-commutative 78.8%

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

      4. mul-1-neg 78.8%

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

      5. unsub-neg 78.8%

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

      6. associate-/l* 78.8%

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

      7. distribute-rgt-neg-in 78.8%

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

      8. distribute-frac-neg 78.8%

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

      9. distribute-neg-in 78.8%

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

      10. metadata-eval 78.8%

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

      11. unsub-neg 78.8%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in alpha around inf 80.4%

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

      1. associate-*r/ 80.4%

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

      2. distribute-lft-in 80.4%

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

      3. metadata-eval 80.4%

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

      4. associate-*r* 80.4%

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

      5. metadata-eval 80.4%

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

      6. *-lft-identity 80.4%

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

   10. Simplified 80.4%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 32000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.55 \cdot 10^{+162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.55e+162) 1.0 (/ beta alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.55e+162) {
		tmp = 1.0;
	} else {
		tmp = beta / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.55d+162) then
        tmp = 1.0d0
    else
        tmp = beta / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.55e+162) {
		tmp = 1.0;
	} else {
		tmp = beta / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.55e+162:
		tmp = 1.0
	else:
		tmp = beta / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.55e+162)
		tmp = 1.0;
	else
		tmp = Float64(beta / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.55e+162)
		tmp = 1.0;
	else
		tmp = beta / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.55e+162], 1.0, N[(beta / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.55e162

   1. Initial program 83.9%

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

      1. +-commutative 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in beta around inf 43.0%

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

   if 1.55e162 < alpha

   1. Initial program 13.8%

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

      1. +-commutative 13.8%

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

   3. Simplified 13.8%

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

   5. Taylor expanded in alpha around inf 86.5%

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

      1. associate--l+ 86.5%

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

      2. sub-neg 86.5%

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

      3. +-commutative 86.5%

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

      4. mul-1-neg 86.5%

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

      5. unsub-neg 86.5%

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

      6. associate-/l* 86.5%

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

      7. distribute-rgt-neg-in 86.5%

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

      8. distribute-frac-neg 86.5%

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

      9. distribute-neg-in 86.5%

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

      10. metadata-eval 86.5%

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

      11. unsub-neg 86.5%

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

   7. Simplified 86.5%

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

   8. Taylor expanded in alpha around inf 92.6%

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

      1. associate-*r/ 92.6%

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

      2. distribute-lft-in 92.6%

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

      3. metadata-eval 92.6%

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

      4. associate-*r* 92.6%

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

      5. metadata-eval 92.6%

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

      6. *-lft-identity 92.6%

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

   10. Simplified 92.6%

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

   11. Taylor expanded in beta around inf 28.9%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.55 \cdot 10^{+162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.1% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 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 beta around inf 38.5%

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

6. Final simplification 38.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))