Octave 3.8, jcobi/1

Percentage Accurate: 74.5% → 99.9%

Time: 10.6s

Alternatives: 10

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: 74.5% 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 - \frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999)
   (/
    (/
     (+
      2.0
      (-
       (* beta (+ 2.0 (- (* (/ beta alpha) -2.0) (/ 6.0 alpha))))
       (/ 4.0 alpha)))
     alpha)
    2.0)
   (/
    (pow
     (pow (- 1.0 (/ (- alpha beta) (+ beta (+ alpha 2.0)))) 3.0)
     0.3333333333333333)
    2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999) {
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0;
	} else {
		tmp = pow(pow((1.0 - ((alpha - beta) / (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) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99999d0)) then
        tmp = ((2.0d0 + ((beta * (2.0d0 + (((beta / alpha) * (-2.0d0)) - (6.0d0 / alpha)))) - (4.0d0 / alpha))) / alpha) / 2.0d0
    else
        tmp = (((1.0d0 - ((alpha - beta) / (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 tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999) {
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0;
	} else {
		tmp = Math.pow(Math.pow((1.0 - ((alpha - beta) / (beta + (alpha + 2.0)))), 3.0), 0.3333333333333333) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999:
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0
	else:
		tmp = math.pow(math.pow((1.0 - ((alpha - beta) / (beta + (alpha + 2.0)))), 3.0), 0.3333333333333333) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * Float64(2.0 + Float64(Float64(Float64(beta / alpha) * -2.0) - Float64(6.0 / alpha)))) - Float64(4.0 / alpha))) / alpha) / 2.0);
	else
		tmp = Float64(((Float64(1.0 - Float64(Float64(alpha - beta) / Float64(beta + Float64(alpha + 2.0)))) ^ 3.0) ^ 0.3333333333333333) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999)
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0;
	else
		tmp = (((1.0 - ((alpha - beta) / (beta + (alpha + 2.0)))) ^ 3.0) ^ 0.3333333333333333) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(N[(2.0 + N[(N[(beta * N[(2.0 + N[(N[(N[(beta / alpha), $MachinePrecision] * -2.0), $MachinePrecision] - N[(6.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Power[N[(1.0 - N[(N[(alpha - beta), $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))) < -0.999990000000000046

   1. Initial program 8.3%

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

      1. +-commutative 8.3%

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

   3. Simplified 8.3%

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

      1. add-cbrt-cube 8.3%

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

      2. pow1/3 8.3%

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

      3. pow3 8.3%

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

      4. associate-+l+ 8.3%

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

   6. Applied egg-rr 8.3%

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

   7. Taylor expanded in alpha around -inf 94.6%

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

      1. associate-*r/ 94.6%

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

   9. Simplified 94.6%

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

   10. Taylor expanded in beta around 0 99.9%

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

       1. associate--l+ 99.9%

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

       2. associate--l+ 99.9%

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

       3. *-commutative 99.9%

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

       4. associate-*r/ 99.9%

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

       5. metadata-eval 99.9%

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

       6. associate-*r/ 99.9%

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

       7. metadata-eval 99.9%

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

   12. Simplified 99.9%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. add-cbrt-cube 99.7%

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

      2. pow1/3 99.8%

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

      3. pow3 99.7%

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

      4. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 - \frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999)
     (/
      (/
       (+
        2.0
        (-
         (* beta (+ 2.0 (- (* (/ beta alpha) -2.0) (/ 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 <= -0.99999) {
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (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 <= (-0.99999d0)) then
        tmp = ((2.0d0 + ((beta * (2.0d0 + (((beta / alpha) * (-2.0d0)) - (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 <= -0.99999) {
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (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 <= -0.99999:
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (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 <= -0.99999)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * Float64(2.0 + Float64(Float64(Float64(beta / alpha) * -2.0) - 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 <= -0.99999)
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (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, -0.99999], N[(N[(N[(2.0 + N[(N[(beta * N[(2.0 + N[(N[(N[(beta / alpha), $MachinePrecision] * -2.0), $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))) < -0.999990000000000046

   1. Initial program 8.3%

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

      1. +-commutative 8.3%

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

   3. Simplified 8.3%

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

      1. add-cbrt-cube 8.3%

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

      2. pow1/3 8.3%

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

      3. pow3 8.3%

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

      4. associate-+l+ 8.3%

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

   6. Applied egg-rr 8.3%

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

   7. Taylor expanded in alpha around -inf 94.6%

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

      1. associate-*r/ 94.6%

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

   9. Simplified 94.6%

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

   10. Taylor expanded in beta around 0 99.9%

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

       1. associate--l+ 99.9%

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

       2. associate--l+ 99.9%

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

       3. *-commutative 99.9%

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

       4. associate-*r/ 99.9%

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

       5. metadata-eval 99.9%

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

       6. associate-*r/ 99.9%

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

       7. metadata-eval 99.9%

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

   12. Simplified 99.9%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99999:\\ \;\;\;\;\frac{\frac{\left(\beta - \left(\frac{4}{\alpha} + \beta \cdot \frac{\beta + 2}{\alpha}\right)\right) + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999)
     (/
      (/
       (+
        (- beta (+ (/ 4.0 alpha) (* beta (/ (+ beta 2.0) alpha))))
        (- 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 <= -0.99999) {
		tmp = (((beta - ((4.0 / alpha) + (beta * ((beta + 2.0) / alpha)))) + (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 <= (-0.99999d0)) then
        tmp = (((beta - ((4.0d0 / alpha) + (beta * ((beta + 2.0d0) / alpha)))) + (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 <= -0.99999) {
		tmp = (((beta - ((4.0 / alpha) + (beta * ((beta + 2.0) / alpha)))) + (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 <= -0.99999:
		tmp = (((beta - ((4.0 / alpha) + (beta * ((beta + 2.0) / alpha)))) + (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 <= -0.99999)
		tmp = Float64(Float64(Float64(Float64(beta - Float64(Float64(4.0 / alpha) + Float64(beta * Float64(Float64(beta + 2.0) / alpha)))) + 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 <= -0.99999)
		tmp = (((beta - ((4.0 / alpha) + (beta * ((beta + 2.0) / alpha)))) + (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, -0.99999], N[(N[(N[(N[(beta - N[(N[(4.0 / alpha), $MachinePrecision] + N[(beta * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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))) < -0.999990000000000046

   1. Initial program 8.3%

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

      1. +-commutative 8.3%

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

   3. Simplified 8.3%

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

      1. add-cbrt-cube 8.3%

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

      2. pow1/3 8.3%

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

      3. pow3 8.3%

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

      4. associate-+l+ 8.3%

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

   6. Applied egg-rr 8.3%

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

   7. Taylor expanded in alpha around -inf 94.6%

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

      1. associate-*r/ 94.6%

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

   9. Simplified 94.6%

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

   10. Taylor expanded in beta around 0 99.3%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\left(\beta - \left(\frac{4}{\alpha} + \beta \cdot \frac{\beta + 2}{\alpha}\right)\right) + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999998) (/ (+ beta 1.0) alpha) (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99999998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.99999998d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99999998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.99999998:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.99999998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.99999998)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99999998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999980000000011

   1. Initial program 7.5%

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

      1. +-commutative 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in alpha around inf 99.0%

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

   6. Taylor expanded in beta around 0 99.0%

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

   7. Taylor expanded in alpha around 0 99.0%

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

   if -0.999999980000000011 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{-70}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq 2.4 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.9e-70)
   (+ 0.5 (* alpha -0.25))
   (if (<= alpha 2.4e+33) 1.0 (/ (+ beta 1.0) alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.9e-70) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= 2.4e+33) {
		tmp = 1.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.9d-70) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else if (alpha <= 2.4d+33) then
        tmp = 1.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.9e-70) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= 2.4e+33) {
		tmp = 1.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.9e-70:
		tmp = 0.5 + (alpha * -0.25)
	elif alpha <= 2.4e+33:
		tmp = 1.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.9e-70)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	elseif (alpha <= 2.4e+33)
		tmp = 1.0;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.9e-70)
		tmp = 0.5 + (alpha * -0.25);
	elseif (alpha <= 2.4e+33)
		tmp = 1.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.9e-70], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[alpha, 2.4e+33], 1.0, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 2.89999999999999971e-70

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow3 100.0%

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

      4. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in beta around 0 78.0%

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

      1. +-commutative 78.0%

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

   9. Simplified 78.0%

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

   10. Taylor expanded in alpha around 0 77.6%

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

       1. *-commutative 77.6%

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

   12. Simplified 77.6%

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

   if 2.89999999999999971e-70 < alpha < 2.4e33

   1. Initial program 96.0%

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

      1. +-commutative 96.0%

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

   3. Simplified 96.0%

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

      1. add-cbrt-cube 96.1%

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

      2. pow1/3 96.1%

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

      3. pow3 96.1%

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

      4. associate-+l+ 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in beta around inf 56.8%

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

   if 2.4e33 < alpha

   1. Initial program 17.2%

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

      1. +-commutative 17.2%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in alpha around inf 89.0%

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

   6. Taylor expanded in beta around 0 89.0%

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

   7. Taylor expanded in alpha around 0 89.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{-70}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq 2.4 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.4e+33)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.4e+33) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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 <= 2.4d+33) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 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 <= 2.4e+33) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.4e+33:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.4e+33)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 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 <= 2.4e+33)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.4e+33], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.4e33

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 95.0%

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

   if 2.4e33 < alpha

   1. Initial program 17.2%

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

      1. +-commutative 17.2%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in alpha around inf 89.0%

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

   6. Taylor expanded in beta around 0 89.0%

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

   7. Taylor expanded in alpha around 0 89.0%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{-70}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq 2.4 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.4e-70)
   (+ 0.5 (* alpha -0.25))
   (if (<= alpha 2.4e+33) 1.0 (/ 1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.4e-70) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= 2.4e+33) {
		tmp = 1.0;
	} 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 <= 1.4d-70) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else if (alpha <= 2.4d+33) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.4e-70) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= 2.4e+33) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.4e-70:
		tmp = 0.5 + (alpha * -0.25)
	elif alpha <= 2.4e+33:
		tmp = 1.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.4e-70)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	elseif (alpha <= 2.4e+33)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.4e-70)
		tmp = 0.5 + (alpha * -0.25);
	elseif (alpha <= 2.4e+33)
		tmp = 1.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.4e-70], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[alpha, 2.4e+33], 1.0, N[(1.0 / alpha), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.4e-70

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow3 100.0%

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

      4. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in beta around 0 78.0%

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

      1. +-commutative 78.0%

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

   9. Simplified 78.0%

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

   10. Taylor expanded in alpha around 0 77.6%

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

       1. *-commutative 77.6%

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

   12. Simplified 77.6%

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

   if 1.4e-70 < alpha < 2.4e33

   1. Initial program 96.0%

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

      1. +-commutative 96.0%

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

   3. Simplified 96.0%

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

      1. add-cbrt-cube 96.1%

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

      2. pow1/3 96.1%

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

      3. pow3 96.1%

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

      4. associate-+l+ 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in beta around inf 56.8%

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

   if 2.4e33 < alpha

   1. Initial program 17.2%

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

      1. +-commutative 17.2%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in alpha around inf 89.0%

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

   6. Taylor expanded in beta around 0 72.6%

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

Alternative 8: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{-70}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 2.5 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2e-70) 0.5 (if (<= alpha 2.5e+33) 1.0 (/ 1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2e-70) {
		tmp = 0.5;
	} else if (alpha <= 2.5e+33) {
		tmp = 1.0;
	} 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 <= 2d-70) then
        tmp = 0.5d0
    else if (alpha <= 2.5d+33) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2e-70) {
		tmp = 0.5;
	} else if (alpha <= 2.5e+33) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2e-70:
		tmp = 0.5
	elif alpha <= 2.5e+33:
		tmp = 1.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2e-70)
		tmp = 0.5;
	elseif (alpha <= 2.5e+33)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2e-70)
		tmp = 0.5;
	elseif (alpha <= 2.5e+33)
		tmp = 1.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2e-70], 0.5, If[LessEqual[alpha, 2.5e+33], 1.0, N[(1.0 / alpha), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.99999999999999999e-70

   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 99.0%

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

   6. Taylor expanded in beta around 0 76.9%

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

   if 1.99999999999999999e-70 < alpha < 2.49999999999999986e33

   1. Initial program 96.0%

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

      1. +-commutative 96.0%

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

   3. Simplified 96.0%

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

      1. add-cbrt-cube 96.1%

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

      2. pow1/3 96.1%

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

      3. pow3 96.1%

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

      4. associate-+l+ 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in beta around inf 56.8%

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

   if 2.49999999999999986e33 < alpha

   1. Initial program 17.2%

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

      1. +-commutative 17.2%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in alpha around inf 89.0%

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

   6. Taylor expanded in beta around 0 72.6%

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

Alternative 9: 71.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 2.05) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.05) {
		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 <= 2.05d0) 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 <= 2.05) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.05:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.05)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.05)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.05], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0499999999999998

   1. Initial program 69.0%

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

      1. +-commutative 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in alpha around 0 64.4%

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

   6. Taylor expanded in beta around 0 63.1%

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

   if 2.0499999999999998 < beta

   1. Initial program 82.2%

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

      1. +-commutative 82.2%

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

   3. Simplified 82.2%

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

      1. add-cbrt-cube 82.2%

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

      2. pow1/3 82.2%

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

      3. pow3 82.2%

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

      4. associate-+l+ 82.2%

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

   6. Applied egg-rr 82.2%

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

   7. Taylor expanded in beta around inf 80.0%

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

Alternative 10: 49.5% 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 72.9%

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

   1. +-commutative 72.9%

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

3. Simplified 72.9%

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

5. Taylor expanded in alpha around 0 69.5%

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

6. Taylor expanded in beta around 0 49.2%

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

Reproduce

herbie shell --seed 2024136 
(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))