Octave 3.8, jcobi/1

Percentage Accurate: 73.5% → 99.8%

Time: 9.9s

Alternatives: 12

Speedup: 1.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{1}{\alpha}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 - \left(\beta \cdot \frac{\beta - -2}{\alpha} + \left(t\_0 + \beta \cdot \left(t\_0 + \left(\frac{\beta}{\alpha} - 2\right)\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left({\left({\left({\left({\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ 1.0 alpha))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9995)
     (/
      (/
       (-
        2.0
        (+
         (* beta (/ (- beta -2.0) alpha))
         (+ t_0 (* beta (+ t_0 (- (/ beta alpha) 2.0))))))
       alpha)
      2.0)
     (pow
      (pow
       (*
        (pow
         (pow
          (pow
           (pow (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 3.0)
           0.3333333333333333)
          3.0)
         0.3333333333333333)
        0.5)
       3.0)
      0.3333333333333333))))

C

double code(double alpha, double beta) {
	double t_0 = 4.0 * (1.0 / alpha);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995) {
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0)))))) / alpha) / 2.0;
	} else {
		tmp = pow(pow((pow(pow(pow(pow((1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))), 3.0), 0.3333333333333333), 3.0), 0.3333333333333333) * 0.5), 3.0), 0.3333333333333333);
	}
	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 = 4.0d0 * (1.0d0 / alpha)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9995d0)) then
        tmp = ((2.0d0 - ((beta * ((beta - (-2.0d0)) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0d0)))))) / alpha) / 2.0d0
    else
        tmp = (((((((1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) ** 3.0d0) ** 0.3333333333333333d0) ** 3.0d0) ** 0.3333333333333333d0) * 0.5d0) ** 3.0d0) ** 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 4.0 * (1.0 / alpha);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995) {
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0)))))) / alpha) / 2.0;
	} else {
		tmp = Math.pow(Math.pow((Math.pow(Math.pow(Math.pow(Math.pow((1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))), 3.0), 0.3333333333333333), 3.0), 0.3333333333333333) * 0.5), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 4.0 * (1.0 / alpha)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995:
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0)))))) / alpha) / 2.0
	else:
		tmp = math.pow(math.pow((math.pow(math.pow(math.pow(math.pow((1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))), 3.0), 0.3333333333333333), 3.0), 0.3333333333333333) * 0.5), 3.0), 0.3333333333333333)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(4.0 * Float64(1.0 / alpha))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9995)
		tmp = Float64(Float64(Float64(2.0 - Float64(Float64(beta * Float64(Float64(beta - -2.0) / alpha)) + Float64(t_0 + Float64(beta * Float64(t_0 + Float64(Float64(beta / alpha) - 2.0)))))) / alpha) / 2.0);
	else
		tmp = (Float64(((((Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) ^ 3.0) ^ 0.3333333333333333) ^ 3.0) ^ 0.3333333333333333) * 0.5) ^ 3.0) ^ 0.3333333333333333;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 4.0 * (1.0 / alpha);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995)
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0)))))) / alpha) / 2.0;
	else
		tmp = (((((((1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) ^ 3.0) ^ 0.3333333333333333) ^ 3.0) ^ 0.3333333333333333) * 0.5) ^ 3.0) ^ 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(4.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9995], N[(N[(N[(2.0 - N[(N[(beta * N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(beta * N[(t$95$0 + N[(N[(beta / alpha), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[Power[N[(N[Power[N[Power[N[Power[N[Power[N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] * 0.5), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99950000000000006

   1. Initial program 7.0%

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

      1. +-commutative 7.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in alpha around inf 94.0%

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

   7. Simplified 94.0%

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

   8. Taylor expanded in beta around 0 99.8%

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

   if -0.99950000000000006 < (/.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. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

      3. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. add-cbrt-cube 99.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2} \cdot \frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}\right) \cdot \frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}}} \]

      2. pow1/3 99.9%

         \[\leadsto \color{blue}{{\left(\left(\frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2} \cdot \frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}\right) \cdot \frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}\right)}^{0.3333333333333333}} \]

      3. pow3 99.9%

         \[\leadsto {\color{blue}{\left({\left(\frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. div-inv 99.9%

         \[\leadsto {\left({\color{blue}{\left(\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1\right) \cdot \frac{1}{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. +-commutative 99.9%

         \[\leadsto {\left({\left(\color{blue}{\left(1 + \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right)\right)} \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. associate-*l/ 99.9%

         \[\leadsto {\left({\left(\left(1 + \color{blue}{\frac{1 \cdot \left(\beta - \alpha\right)}{\beta + \left(\alpha + 2\right)}}\right) \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      7. *-un-lft-identity 99.9%

         \[\leadsto {\left({\left(\left(1 + \frac{\color{blue}{\beta - \alpha}}{\beta + \left(\alpha + 2\right)}\right) \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      8. metadata-eval 99.9%

         \[\leadsto {\left({\left(\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \color{blue}{0.5}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left({\left(\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 99.9%

         \[\leadsto {\left({\left(\color{blue}{\sqrt[3]{\left(\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right) \cdot \left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]

      2. pow1/3 99.9%

         \[\leadsto {\left({\left(\color{blue}{{\left(\left(\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right) \cdot \left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right)}^{0.3333333333333333}} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]

      3. pow3 99.9%

         \[\leadsto {\left({\left({\color{blue}{\left({\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}}^{0.3333333333333333} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]

   10. Applied egg-rr 99.9%

       \[\leadsto {\left({\left(\color{blue}{{\left({\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}^{0.3333333333333333}} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]
   11. Step-by-step derivation

       1. add-cbrt-cube 99.9%

          \[\leadsto {\left({\left(\color{blue}{\sqrt[3]{\left(\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right) \cdot \left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]

       2. pow1/3 99.9%

          \[\leadsto {\left({\left(\color{blue}{{\left(\left(\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right) \cdot \left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right)}^{0.3333333333333333}} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]

       3. pow3 99.9%

          \[\leadsto {\left({\left({\color{blue}{\left({\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}}^{0.3333333333333333} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]

   12. Applied egg-rr 99.9%

       \[\leadsto {\left({\left({\left({\color{blue}{\left({\left({\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 - \left(\beta \cdot \frac{\beta - -2}{\alpha} + \left(4 \cdot \frac{1}{\alpha} + \beta \cdot \left(4 \cdot \frac{1}{\alpha} + \left(\frac{\beta}{\alpha} - 2\right)\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left({\left({\left({\left({\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{1}{\alpha}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 - \left(\beta \cdot \frac{\beta - -2}{\alpha} + \left(t\_0 + \beta \cdot \left(t\_0 + \left(\frac{\beta}{\alpha} - 2\right)\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left({\left({\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}^{0.3333333333333333} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ 1.0 alpha))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9995)
     (/
      (/
       (-
        2.0
        (+
         (* beta (/ (- beta -2.0) alpha))
         (+ t_0 (* beta (+ t_0 (- (/ beta alpha) 2.0))))))
       alpha)
      2.0)
     (pow
      (pow
       (*
        (pow
         (pow (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 3.0)
         0.3333333333333333)
        0.5)
       3.0)
      0.3333333333333333))))

C

double code(double alpha, double beta) {
	double t_0 = 4.0 * (1.0 / alpha);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995) {
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0)))))) / alpha) / 2.0;
	} else {
		tmp = pow(pow((pow(pow((1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))), 3.0), 0.3333333333333333) * 0.5), 3.0), 0.3333333333333333);
	}
	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 = 4.0d0 * (1.0d0 / alpha)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9995d0)) then
        tmp = ((2.0d0 - ((beta * ((beta - (-2.0d0)) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0d0)))))) / alpha) / 2.0d0
    else
        tmp = (((((1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) ** 3.0d0) ** 0.3333333333333333d0) * 0.5d0) ** 3.0d0) ** 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 4.0 * (1.0 / alpha);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995) {
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0)))))) / alpha) / 2.0;
	} else {
		tmp = Math.pow(Math.pow((Math.pow(Math.pow((1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))), 3.0), 0.3333333333333333) * 0.5), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 4.0 * (1.0 / alpha)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995:
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0)))))) / alpha) / 2.0
	else:
		tmp = math.pow(math.pow((math.pow(math.pow((1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))), 3.0), 0.3333333333333333) * 0.5), 3.0), 0.3333333333333333)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(4.0 * Float64(1.0 / alpha))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9995)
		tmp = Float64(Float64(Float64(2.0 - Float64(Float64(beta * Float64(Float64(beta - -2.0) / alpha)) + Float64(t_0 + Float64(beta * Float64(t_0 + Float64(Float64(beta / alpha) - 2.0)))))) / alpha) / 2.0);
	else
		tmp = (Float64(((Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) ^ 3.0) ^ 0.3333333333333333) * 0.5) ^ 3.0) ^ 0.3333333333333333;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 4.0 * (1.0 / alpha);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995)
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0)))))) / alpha) / 2.0;
	else
		tmp = (((((1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) ^ 3.0) ^ 0.3333333333333333) * 0.5) ^ 3.0) ^ 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(4.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9995], N[(N[(N[(2.0 - N[(N[(beta * N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(beta * N[(t$95$0 + N[(N[(beta / alpha), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[Power[N[(N[Power[N[Power[N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] * 0.5), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99950000000000006

   1. Initial program 7.0%

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

      1. +-commutative 7.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in alpha around inf 94.0%

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

   7. Simplified 94.0%

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

   8. Taylor expanded in beta around 0 99.8%

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

   if -0.99950000000000006 < (/.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. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

      3. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. add-cbrt-cube 99.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2} \cdot \frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}\right) \cdot \frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}}} \]

      2. pow1/3 99.9%

         \[\leadsto \color{blue}{{\left(\left(\frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2} \cdot \frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}\right) \cdot \frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}\right)}^{0.3333333333333333}} \]

      3. pow3 99.9%

         \[\leadsto {\color{blue}{\left({\left(\frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. div-inv 99.9%

         \[\leadsto {\left({\color{blue}{\left(\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1\right) \cdot \frac{1}{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. +-commutative 99.9%

         \[\leadsto {\left({\left(\color{blue}{\left(1 + \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right)\right)} \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. associate-*l/ 99.9%

         \[\leadsto {\left({\left(\left(1 + \color{blue}{\frac{1 \cdot \left(\beta - \alpha\right)}{\beta + \left(\alpha + 2\right)}}\right) \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      7. *-un-lft-identity 99.9%

         \[\leadsto {\left({\left(\left(1 + \frac{\color{blue}{\beta - \alpha}}{\beta + \left(\alpha + 2\right)}\right) \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      8. metadata-eval 99.9%

         \[\leadsto {\left({\left(\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \color{blue}{0.5}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left({\left(\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 99.9%

         \[\leadsto {\left({\left(\color{blue}{\sqrt[3]{\left(\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right) \cdot \left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]

      2. pow1/3 99.9%

         \[\leadsto {\left({\left(\color{blue}{{\left(\left(\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right) \cdot \left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right)}^{0.3333333333333333}} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]

      3. pow3 99.9%

         \[\leadsto {\left({\left({\color{blue}{\left({\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}}^{0.3333333333333333} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]

   10. Applied egg-rr 99.9%

       \[\leadsto {\left({\left(\color{blue}{{\left({\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}^{0.3333333333333333}} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 - \left(\beta \cdot \frac{\beta - -2}{\alpha} + \left(4 \cdot \frac{1}{\alpha} + \beta \cdot \left(4 \cdot \frac{1}{\alpha} + \left(\frac{\beta}{\alpha} - 2\right)\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left({\left({\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}^{0.3333333333333333} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{1}{\alpha}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 - \left(\beta \cdot \frac{\beta - -2}{\alpha} + \left(t\_0 + \beta \cdot \left(t\_0 + \left(\frac{\beta}{\alpha} - 2\right)\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ 1.0 alpha))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9995)
     (/
      (/
       (-
        2.0
        (+
         (* beta (/ (- beta -2.0) alpha))
         (+ t_0 (* beta (+ t_0 (- (/ beta alpha) 2.0))))))
       alpha)
      2.0)
     (/
      (pow
       (pow (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 3.0)
       0.3333333333333333)
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = 4.0 * (1.0 / alpha);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995) {
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0)))))) / alpha) / 2.0;
	} else {
		tmp = pow(pow((1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))), 3.0), 0.3333333333333333) / 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 = 4.0d0 * (1.0d0 / alpha)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9995d0)) then
        tmp = ((2.0d0 - ((beta * ((beta - (-2.0d0)) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0d0)))))) / alpha) / 2.0d0
    else
        tmp = (((1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) ** 3.0d0) ** 0.3333333333333333d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 4.0 * (1.0 / alpha);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995) {
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0)))))) / alpha) / 2.0;
	} else {
		tmp = Math.pow(Math.pow((1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))), 3.0), 0.3333333333333333) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 4.0 * (1.0 / alpha)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995:
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0)))))) / alpha) / 2.0
	else:
		tmp = math.pow(math.pow((1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))), 3.0), 0.3333333333333333) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(4.0 * Float64(1.0 / alpha))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9995)
		tmp = Float64(Float64(Float64(2.0 - Float64(Float64(beta * Float64(Float64(beta - -2.0) / alpha)) + Float64(t_0 + Float64(beta * Float64(t_0 + Float64(Float64(beta / alpha) - 2.0)))))) / alpha) / 2.0);
	else
		tmp = Float64(((Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) ^ 3.0) ^ 0.3333333333333333) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 4.0 * (1.0 / alpha);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995)
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_0 + (beta * (t_0 + ((beta / alpha) - 2.0)))))) / alpha) / 2.0;
	else
		tmp = (((1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) ^ 3.0) ^ 0.3333333333333333) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(4.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9995], N[(N[(N[(2.0 - N[(N[(beta * N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(beta * N[(t$95$0 + N[(N[(beta / alpha), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Power[N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $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))) < -0.99950000000000006

   1. Initial program 7.0%

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

      1. +-commutative 7.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in alpha around inf 94.0%

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

   7. Simplified 94.0%

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

   8. Taylor expanded in beta around 0 99.8%

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

   if -0.99950000000000006 < (/.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. add-cbrt-cube 99.9%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1\right) \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1\right)\right) \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1\right)}}}{2} \]

      2. pow1/3 99.9%

         \[\leadsto \frac{\color{blue}{{\left(\left(\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1\right) \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1\right)\right) \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1\right)\right)}^{0.3333333333333333}}}{2} \]

      3. pow3 99.9%

         \[\leadsto \frac{{\color{blue}{\left({\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1\right)}^{3}\right)}}^{0.3333333333333333}}{2} \]

      4. associate-+l+ 99.9%

         \[\leadsto \frac{{\left({\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}} + 1\right)}^{3}\right)}^{0.3333333333333333}}{2} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{{\left({\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1\right)}^{3}\right)}^{0.3333333333333333}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 - \left(\beta \cdot \frac{\beta - -2}{\alpha} + \left(4 \cdot \frac{1}{\alpha} + \beta \cdot \left(4 \cdot \frac{1}{\alpha} + \left(\frac{\beta}{\alpha} - 2\right)\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ t_1 := 4 \cdot \frac{1}{\alpha}\\ \mathbf{if}\;t\_0 \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 - \left(\beta \cdot \frac{\beta - -2}{\alpha} + \left(t\_1 + \beta \cdot \left(t\_1 + \left(\frac{\beta}{\alpha} - 2\right)\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0)))
        (t_1 (* 4.0 (/ 1.0 alpha))))
   (if (<= t_0 -0.9995)
     (/
      (/
       (-
        2.0
        (+
         (* beta (/ (- beta -2.0) alpha))
         (+ t_1 (* beta (+ t_1 (- (/ beta alpha) 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 t_1 = 4.0 * (1.0 / alpha);
	double tmp;
	if (t_0 <= -0.9995) {
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_1 + (beta * (t_1 + ((beta / alpha) - 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) :: t_1
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    t_1 = 4.0d0 * (1.0d0 / alpha)
    if (t_0 <= (-0.9995d0)) then
        tmp = ((2.0d0 - ((beta * ((beta - (-2.0d0)) / alpha)) + (t_1 + (beta * (t_1 + ((beta / alpha) - 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 t_1 = 4.0 * (1.0 / alpha);
	double tmp;
	if (t_0 <= -0.9995) {
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_1 + (beta * (t_1 + ((beta / alpha) - 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)
	t_1 = 4.0 * (1.0 / alpha)
	tmp = 0
	if t_0 <= -0.9995:
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_1 + (beta * (t_1 + ((beta / alpha) - 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))
	t_1 = Float64(4.0 * Float64(1.0 / alpha))
	tmp = 0.0
	if (t_0 <= -0.9995)
		tmp = Float64(Float64(Float64(2.0 - Float64(Float64(beta * Float64(Float64(beta - -2.0) / alpha)) + Float64(t_1 + Float64(beta * Float64(t_1 + Float64(Float64(beta / alpha) - 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);
	t_1 = 4.0 * (1.0 / alpha);
	tmp = 0.0;
	if (t_0 <= -0.9995)
		tmp = ((2.0 - ((beta * ((beta - -2.0) / alpha)) + (t_1 + (beta * (t_1 + ((beta / alpha) - 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]}, Block[{t$95$1 = N[(4.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9995], N[(N[(N[(2.0 - N[(N[(beta * N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(beta * N[(t$95$1 + N[(N[(beta / alpha), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $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))) < -0.99950000000000006

   1. Initial program 7.0%

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

      1. +-commutative 7.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in alpha around inf 94.0%

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

   7. Simplified 94.0%

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

   8. Taylor expanded in beta around 0 99.8%

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

   if -0.99950000000000006 < (/.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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 - \left(\beta \cdot \frac{\beta - -2}{\alpha} + \left(4 \cdot \frac{1}{\alpha} + \beta \cdot \left(4 \cdot \frac{1}{\alpha} + \left(\frac{\beta}{\alpha} - 2\right)\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9999995:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9999995)
     (/ (/ (+ 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 <= -0.9999995) {
		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 <= (-0.9999995d0)) 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 <= -0.9999995) {
		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 <= -0.9999995:
		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 <= -0.9999995)
		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 <= -0.9999995)
		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, -0.9999995], 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))) < -0.999999500000000041

   1. Initial program 6.2%

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

      1. +-commutative 6.2%

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

   3. Simplified 6.2%

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

   5. Taylor expanded in alpha around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. distribute-neg-frac2 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

      6. +-commutative 99.6%

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

      7. mul-1-neg 99.6%

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

      8. unsub-neg 99.6%

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

   7. Simplified 99.6%

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

   if -0.999999500000000041 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999995:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.2e+16)
   (/ (+ 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 <= 2.2e+16) {
		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 <= 2.2d+16) 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 <= 2.2e+16) {
		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 <= 2.2e+16:
		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 <= 2.2e+16)
		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 <= 2.2e+16)
		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, 2.2e+16], 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 < 2.2e16

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 97.9%

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

   if 2.2e16 < alpha

   1. Initial program 16.9%

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

      1. +-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in alpha around -inf 89.2%

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

      1. mul-1-neg 89.2%

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

      2. distribute-neg-frac2 89.2%

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

      3. associate--r+ 89.2%

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

      4. sub-neg 89.2%

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

      5. metadata-eval 89.2%

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

      6. +-commutative 89.2%

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

      7. mul-1-neg 89.2%

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

      8. unsub-neg 89.2%

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

   7. Simplified 89.2%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.15e+17)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.15e+17) {
		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 <= 2.15d+17) 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 <= 2.15e+17) {
		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 <= 2.15e+17:
		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 <= 2.15e+17)
		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 <= 2.15e+17)
		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, 2.15e+17], 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 < 2.15e17

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 97.9%

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

   if 2.15e17 < alpha

   1. Initial program 16.9%

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

      1. +-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in alpha around -inf 89.2%

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

      1. mul-1-neg 89.2%

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

      2. distribute-neg-frac2 89.2%

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

      3. associate--r+ 89.2%

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

      4. sub-neg 89.2%

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

      5. metadata-eval 89.2%

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

      6. +-commutative 89.2%

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

      7. mul-1-neg 89.2%

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

      8. unsub-neg 89.2%

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

   7. Simplified 89.2%

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

   8. Taylor expanded in beta around 0 89.2%

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

      1. +-commutative 89.2%

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

   10. Simplified 89.2%

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

   11. Taylor expanded in alpha around 0 89.2%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.0) (+ 0.5 (* alpha -0.25)) (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.0) {
		tmp = 0.5 + (alpha * -0.25);
	} 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 <= 2.0d0) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    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 <= 2.0) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.0:
		tmp = 0.5 + (alpha * -0.25)
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.0], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], N[(N[(beta + 1.0), $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 65.6%

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

      1. +-commutative 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in alpha around 0 65.2%

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

      1. *-commutative 65.2%

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

   10. Simplified 65.2%

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

   if 2 < alpha

   1. Initial program 19.9%

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

      1. +-commutative 19.9%

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

   3. Simplified 19.9%

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

   5. Taylor expanded in alpha around -inf 86.6%

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

      1. mul-1-neg 86.6%

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

      2. distribute-neg-frac2 86.6%

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

      3. associate--r+ 86.6%

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

      4. sub-neg 86.6%

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

      5. metadata-eval 86.6%

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

      6. +-commutative 86.6%

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

      7. mul-1-neg 86.6%

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

      8. unsub-neg 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in beta around 0 86.6%

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

      1. +-commutative 86.6%

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

   10. Simplified 86.6%

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

   11. Taylor expanded in alpha around 0 86.6%

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

4. Final simplification 73.6%

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

Alternative 9: 69.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.95:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.95) (+ 0.5 (* alpha -0.25)) (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.95) {
		tmp = 0.5 + (alpha * -0.25);
	} 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 <= 0.95d0) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.95) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.95:
		tmp = 0.5 + (alpha * -0.25)
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.95)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.95)
		tmp = 0.5 + (alpha * -0.25);
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.95], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.94999999999999996

   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 65.6%

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

      1. +-commutative 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in alpha around 0 65.2%

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

      1. *-commutative 65.2%

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

   10. Simplified 65.2%

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

   if 0.94999999999999996 < alpha

   1. Initial program 19.9%

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

      1. +-commutative 19.9%

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

   3. Simplified 19.9%

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

   5. Taylor expanded in beta around 0 6.9%

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

      1. +-commutative 6.9%

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

   7. Simplified 6.9%

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

   8. Taylor expanded in alpha around inf 69.8%

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

Alternative 10: 69.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.9:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= alpha 1.9) 0.5 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.9) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.9d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.9) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.9:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.9)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.9)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.9], 0.5, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.8999999999999999

   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 65.6%

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

      1. +-commutative 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in alpha around 0 64.7%

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

   if 1.8999999999999999 < alpha

   1. Initial program 19.9%

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

      1. +-commutative 19.9%

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

   3. Simplified 19.9%

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

   5. Taylor expanded in beta around 0 6.9%

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

      1. +-commutative 6.9%

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

   7. Simplified 6.9%

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

   8. Taylor expanded in alpha around inf 69.8%

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

Alternative 11: 70.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 2.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 60.3%

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

      1. +-commutative 60.3%

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

   3. Simplified 60.3%

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

   5. Taylor expanded in beta around 0 59.3%

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

      1. +-commutative 59.3%

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

   7. Simplified 59.3%

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

   8. Taylor expanded in alpha around 0 57.1%

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

   if 2 < beta

   1. Initial program 83.2%

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

      1. +-commutative 83.2%

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

   3. Simplified 83.2%

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

      1. clear-num 83.2%

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

      2. associate-/r/ 83.2%

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

      3. associate-+l+ 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. add-cbrt-cube 83.3%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2} \cdot \frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}\right) \cdot \frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}}} \]

      2. pow1/3 83.2%

         \[\leadsto \color{blue}{{\left(\left(\frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2} \cdot \frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}\right) \cdot \frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}\right)}^{0.3333333333333333}} \]

      3. pow3 83.2%

         \[\leadsto {\color{blue}{\left({\left(\frac{\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1}{2}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. div-inv 83.2%

         \[\leadsto {\left({\color{blue}{\left(\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right) + 1\right) \cdot \frac{1}{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. +-commutative 83.2%

         \[\leadsto {\left({\left(\color{blue}{\left(1 + \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(\beta - \alpha\right)\right)} \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. associate-*l/ 83.2%

         \[\leadsto {\left({\left(\left(1 + \color{blue}{\frac{1 \cdot \left(\beta - \alpha\right)}{\beta + \left(\alpha + 2\right)}}\right) \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      7. *-un-lft-identity 83.2%

         \[\leadsto {\left({\left(\left(1 + \frac{\color{blue}{\beta - \alpha}}{\beta + \left(\alpha + 2\right)}\right) \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      8. metadata-eval 83.2%

         \[\leadsto {\left({\left(\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \color{blue}{0.5}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{{\left({\left(\left(1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333}} \]

   9. Taylor expanded in beta around inf 80.3%

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

Alternative 12: 48.8% 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 68.7%

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

   1. +-commutative 68.7%

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

3. Simplified 68.7%

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

5. Taylor expanded in beta around 0 42.7%

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

   1. +-commutative 42.7%

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

7. Simplified 42.7%

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

8. Taylor expanded in alpha around 0 42.2%

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

Reproduce

herbie shell --seed 2024131 
(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))