Octave 3.8, jcobi/1

Percentage Accurate: 74.6% → 99.9%

Time: 24.3s

Alternatives: 9

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: 74.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot \left(2 + \left(-2 \cdot \frac{\beta}{\alpha} - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999995)
     (/
      (/
       (+
        2.0
        (-
         (* beta (+ 2.0 (- (* -2.0 (/ beta alpha)) (/ 6.0 alpha))))
         (/ 4.0 alpha)))
       alpha)
      2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999995) {
		tmp = ((2.0 + ((beta * (2.0 + ((-2.0 * (beta / alpha)) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.999995d0)) then
        tmp = ((2.0d0 + ((beta * (2.0d0 + (((-2.0d0) * (beta / alpha)) - (6.0d0 / alpha)))) - (4.0d0 / alpha))) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999995) {
		tmp = ((2.0 + ((beta * (2.0 + ((-2.0 * (beta / alpha)) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.999995:
		tmp = ((2.0 + ((beta * (2.0 + ((-2.0 * (beta / alpha)) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.999995)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * Float64(2.0 + Float64(Float64(-2.0 * Float64(beta / alpha)) - Float64(6.0 / alpha)))) - Float64(4.0 / alpha))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.999995)
		tmp = ((2.0 + ((beta * (2.0 + ((-2.0 * (beta / alpha)) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.999995], N[(N[(N[(2.0 + N[(N[(beta * N[(2.0 + N[(N[(-2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] - N[(6.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99999499999999997

   1. Initial program 8.2%

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

      1. +-commutative 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in alpha around inf 95.5%

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

      1. +-commutative 95.5%

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

      2. associate--l+ 95.5%

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

      3. +-commutative 95.5%

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

      4. mul-1-neg 95.5%

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

      5. unsub-neg 95.5%

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

      6. sub-neg 95.5%

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

      7. associate-/l* 95.5%

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

      8. distribute-rgt-neg-in 95.5%

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

      9. distribute-frac-neg 95.5%

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

      10. distribute-neg-in 95.5%

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

      11. metadata-eval 95.5%

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

      12. unsub-neg 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in beta around 0 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate--l+ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

   10. Simplified 99.6%

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

   if -0.99999499999999997 < (/.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. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot \left(2 + \left(-2 \cdot \frac{\beta}{\alpha} - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.999995:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 2 - \frac{4}{\alpha}\right) + \left(2 + \beta \cdot \frac{-2 - \beta}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999995)
     (/
      (/
       (+
        (- (* beta 2.0) (/ 4.0 alpha))
        (+ 2.0 (* beta (/ (- -2.0 beta) alpha))))
       alpha)
      2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999995) {
		tmp = ((((beta * 2.0) - (4.0 / alpha)) + (2.0 + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.999995d0)) then
        tmp = ((((beta * 2.0d0) - (4.0d0 / alpha)) + (2.0d0 + (beta * (((-2.0d0) - beta) / alpha)))) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999995) {
		tmp = ((((beta * 2.0) - (4.0 / alpha)) + (2.0 + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.999995:
		tmp = ((((beta * 2.0) - (4.0 / alpha)) + (2.0 + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.999995)
		tmp = Float64(Float64(Float64(Float64(Float64(beta * 2.0) - Float64(4.0 / alpha)) + Float64(2.0 + Float64(beta * Float64(Float64(-2.0 - beta) / alpha)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.999995)
		tmp = ((((beta * 2.0) - (4.0 / alpha)) + (2.0 + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.999995], N[(N[(N[(N[(N[(beta * 2.0), $MachinePrecision] - N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(beta * N[(N[(-2.0 - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99999499999999997

   1. Initial program 8.2%

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

      1. +-commutative 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in alpha around inf 95.5%

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

      1. +-commutative 95.5%

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

      2. associate--l+ 95.5%

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

      3. +-commutative 95.5%

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

      4. mul-1-neg 95.5%

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

      5. unsub-neg 95.5%

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

      6. sub-neg 95.5%

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

      7. associate-/l* 95.5%

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

      8. distribute-rgt-neg-in 95.5%

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

      9. distribute-frac-neg 95.5%

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

      10. distribute-neg-in 95.5%

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

      11. metadata-eval 95.5%

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

      12. unsub-neg 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in beta around 0 99.4%

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

   if -0.99999499999999997 < (/.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. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 2 - \frac{4}{\alpha}\right) + \left(2 + \beta \cdot \frac{-2 - \beta}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9999998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9999998)
     (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9999998) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.9999998d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9999998) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.9999998:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.9999998)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.9999998)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9999998], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999799999999994

   1. Initial program 6.4%

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

      1. +-commutative 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in alpha around -inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. sub-neg 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. neg-mul-1 99.4%

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

      5. mul-1-neg 99.4%

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

      6. remove-double-neg 99.4%

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

      7. neg-mul-1 99.4%

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

      8. remove-double-neg 99.4%

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

   7. Simplified 99.4%

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

   if -0.999999799999999994 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 8.0)
   (/ (+ 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 <= 8.0) {
		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 <= 8.0d0) 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 <= 8.0) {
		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 <= 8.0:
		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 <= 8.0)
		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 <= 8.0)
		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, 8.0], 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 < 8

   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 98.7%

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

   if 8 < alpha

   1. Initial program 21.7%

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

      1. +-commutative 21.7%

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

   3. Simplified 21.7%

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

   5. Taylor expanded in alpha around -inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. sub-neg 85.0%

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

      3. distribute-lft-in 85.0%

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

      4. neg-mul-1 85.0%

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

      5. mul-1-neg 85.0%

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

      6. remove-double-neg 85.0%

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

      7. neg-mul-1 85.0%

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

      8. remove-double-neg 85.0%

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

   7. Simplified 85.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{2}{\alpha}}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 8.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (- 1.0 (/ 2.0 alpha)) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (1.0 - (2.0 / alpha)) / 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 <= 8.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (1.0d0 - (2.0d0 / alpha)) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (1.0 - (2.0 / alpha)) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 8.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (1.0 - (2.0 / alpha)) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 8.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(2.0 / alpha)) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 8.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (1.0 - (2.0 / alpha)) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 8.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8

   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 98.7%

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

   if 8 < alpha

   1. Initial program 21.7%

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

      1. +-commutative 21.7%

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

   3. Simplified 21.7%

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

   5. Taylor expanded in alpha around inf 81.6%

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

      1. +-commutative 81.6%

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

      2. associate--l+ 81.6%

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

      3. +-commutative 81.6%

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

      4. mul-1-neg 81.6%

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

      5. unsub-neg 81.6%

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

      6. sub-neg 81.6%

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

      7. associate-/l* 81.6%

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

      8. distribute-rgt-neg-in 81.6%

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

      9. distribute-frac-neg 81.6%

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

      10. distribute-neg-in 81.6%

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

      11. metadata-eval 81.6%

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

      12. unsub-neg 81.6%

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

   7. Simplified 81.6%

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

   8. Taylor expanded in beta around 0 73.4%

      \[\leadsto \frac{\color{blue}{\frac{2 - 4 \cdot \frac{1}{\alpha}}{\alpha}}}{2} \]
   9. Step-by-step derivation

      1. associate-*r/ 73.4%

         \[\leadsto \frac{\frac{2 - \color{blue}{\frac{4 \cdot 1}{\alpha}}}{\alpha}}{2} \]

      2. metadata-eval 73.4%

         \[\leadsto \frac{\frac{2 - \frac{\color{blue}{4}}{\alpha}}{\alpha}}{2} \]

   10. Simplified 73.4%

       \[\leadsto \frac{\color{blue}{\frac{2 - \frac{4}{\alpha}}{\alpha}}}{2} \]

   11. Taylor expanded in alpha around inf 73.4%

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

       1. associate-*r/ 73.4%

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

       2. metadata-eval 73.4%

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

   13. Simplified 73.4%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{2}{\alpha}}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.0) {
		tmp = (1.0 + (alpha * -0.5)) / 2.0;
	} else {
		tmp = (1.0 - (2.0 / alpha)) / 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 <= 2.0d0) then
        tmp = (1.0d0 + (alpha * (-0.5d0))) / 2.0d0
    else
        tmp = (1.0d0 - (2.0d0 / alpha)) / alpha
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 2

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 74.3%

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

      1. +-commutative 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in alpha around 0 73.6%

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

      1. *-commutative 73.6%

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

   10. Simplified 73.6%

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

   if 2 < alpha

   1. Initial program 21.7%

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

      1. +-commutative 21.7%

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

   3. Simplified 21.7%

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

   5. Taylor expanded in alpha around inf 81.6%

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

      1. +-commutative 81.6%

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

      2. associate--l+ 81.6%

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

      3. +-commutative 81.6%

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

      4. mul-1-neg 81.6%

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

      5. unsub-neg 81.6%

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

      6. sub-neg 81.6%

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

      7. associate-/l* 81.6%

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

      8. distribute-rgt-neg-in 81.6%

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

      9. distribute-frac-neg 81.6%

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

      10. distribute-neg-in 81.6%

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

      11. metadata-eval 81.6%

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

      12. unsub-neg 81.6%

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

   7. Simplified 81.6%

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

   8. Taylor expanded in beta around 0 73.4%

      \[\leadsto \frac{\color{blue}{\frac{2 - 4 \cdot \frac{1}{\alpha}}{\alpha}}}{2} \]
   9. Step-by-step derivation

      1. associate-*r/ 73.4%

         \[\leadsto \frac{\frac{2 - \color{blue}{\frac{4 \cdot 1}{\alpha}}}{\alpha}}{2} \]

      2. metadata-eval 73.4%

         \[\leadsto \frac{\frac{2 - \frac{\color{blue}{4}}{\alpha}}{\alpha}}{2} \]

   10. Simplified 73.4%

       \[\leadsto \frac{\color{blue}{\frac{2 - \frac{4}{\alpha}}{\alpha}}}{2} \]

   11. Taylor expanded in alpha around inf 73.4%

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

       1. associate-*r/ 73.4%

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

       2. metadata-eval 73.4%

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

   13. Simplified 73.4%

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

Alternative 7: 69.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.15:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.15) (/ (+ 1.0 (* alpha -0.5)) 2.0) (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.15) {
		tmp = (1.0 + (alpha * -0.5)) / 2.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.15d0) then
        tmp = (1.0d0 + (alpha * (-0.5d0))) / 2.0d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.15) {
		tmp = (1.0 + (alpha * -0.5)) / 2.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.15:
		tmp = (1.0 + (alpha * -0.5)) / 2.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.15)
		tmp = Float64(Float64(1.0 + Float64(alpha * -0.5)) / 2.0);
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.15)
		tmp = (1.0 + (alpha * -0.5)) / 2.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.15], N[(N[(1.0 + N[(alpha * -0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.1499999999999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 74.3%

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

      1. +-commutative 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in alpha around 0 73.6%

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

      1. *-commutative 73.6%

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

   10. Simplified 73.6%

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

   if 1.1499999999999999 < alpha

   1. Initial program 21.7%

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

      1. +-commutative 21.7%

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

   3. Simplified 21.7%

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

   5. Taylor expanded in alpha around inf 81.6%

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

      1. +-commutative 81.6%

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

      2. associate--l+ 81.6%

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

      3. +-commutative 81.6%

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

      4. mul-1-neg 81.6%

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

      5. unsub-neg 81.6%

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

      6. sub-neg 81.6%

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

      7. associate-/l* 81.6%

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

      8. distribute-rgt-neg-in 81.6%

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

      9. distribute-frac-neg 81.6%

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

      10. distribute-neg-in 81.6%

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

      11. metadata-eval 81.6%

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

      12. unsub-neg 81.6%

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

   7. Simplified 81.6%

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

   8. Taylor expanded in beta around 0 73.4%

      \[\leadsto \frac{\color{blue}{\frac{2 - 4 \cdot \frac{1}{\alpha}}{\alpha}}}{2} \]
   9. Step-by-step derivation

      1. associate-*r/ 73.4%

         \[\leadsto \frac{\frac{2 - \color{blue}{\frac{4 \cdot 1}{\alpha}}}{\alpha}}{2} \]

      2. metadata-eval 73.4%

         \[\leadsto \frac{\frac{2 - \frac{\color{blue}{4}}{\alpha}}{\alpha}}{2} \]

   10. Simplified 73.4%

       \[\leadsto \frac{\color{blue}{\frac{2 - \frac{4}{\alpha}}{\alpha}}}{2} \]

   11. Taylor expanded in alpha around inf 72.2%

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

Alternative 8: 68.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= alpha 2.3) 0.5 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.3) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.3d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.3) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.3:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.3)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.3)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.3], 0.5, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.2999999999999998

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 74.3%

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

      1. +-commutative 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in alpha around 0 73.0%

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

   if 2.2999999999999998 < alpha

   1. Initial program 21.7%

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

      1. +-commutative 21.7%

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

   3. Simplified 21.7%

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

   5. Taylor expanded in alpha around inf 81.6%

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

      1. +-commutative 81.6%

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

      2. associate--l+ 81.6%

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

      3. +-commutative 81.6%

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

      4. mul-1-neg 81.6%

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

      5. unsub-neg 81.6%

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

      6. sub-neg 81.6%

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

      7. associate-/l* 81.6%

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

      8. distribute-rgt-neg-in 81.6%

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

      9. distribute-frac-neg 81.6%

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

      10. distribute-neg-in 81.6%

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

      11. metadata-eval 81.6%

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

      12. unsub-neg 81.6%

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

   7. Simplified 81.6%

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

   8. Taylor expanded in beta around 0 73.4%

      \[\leadsto \frac{\color{blue}{\frac{2 - 4 \cdot \frac{1}{\alpha}}{\alpha}}}{2} \]
   9. Step-by-step derivation

      1. associate-*r/ 73.4%

         \[\leadsto \frac{\frac{2 - \color{blue}{\frac{4 \cdot 1}{\alpha}}}{\alpha}}{2} \]

      2. metadata-eval 73.4%

         \[\leadsto \frac{\frac{2 - \frac{\color{blue}{4}}{\alpha}}{\alpha}}{2} \]

   10. Simplified 73.4%

       \[\leadsto \frac{\color{blue}{\frac{2 - \frac{4}{\alpha}}{\alpha}}}{2} \]

   11. Taylor expanded in alpha around inf 72.2%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.4% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 73.4%

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

   1. +-commutative 73.4%

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

3. Simplified 73.4%

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

5. Taylor expanded in beta around 0 51.9%

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

   1. +-commutative 51.9%

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

7. Simplified 51.9%

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

8. Taylor expanded in alpha around 0 50.7%

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

9. Final simplification 50.7%

   \[\leadsto 0.5 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024086 
(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))