Octave 3.8, jcobi/1

Percentage Accurate: 75.4% → 99.9%

Time: 15.3s

Alternatives: 15

Speedup: 1.2×

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: 75.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\beta\right) - \left(\beta + 2\right)\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;\beta \leq 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{\left(0.5 \cdot \frac{t_1 - -2 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\beta + 2\right)\right)\right)}{t_1} - 0.5\right) - \frac{\alpha}{t_0}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{0.5 + \frac{0.5 \cdot \alpha}{\beta}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (- beta) (+ beta 2.0))) (t_1 (pow t_0 2.0)))
   (if (<= beta 1e+15)
     (/
      (/
       1.0
       (-
        (-
         (*
          0.5
          (/ (- t_1 (* -2.0 (* (+ beta 2.0) (+ beta (+ beta 2.0))))) t_1))
         0.5)
        (/ alpha t_0)))
      2.0)
     (/ (/ 1.0 (+ 0.5 (/ (* 0.5 alpha) beta))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = -beta - (beta + 2.0);
	double t_1 = pow(t_0, 2.0);
	double tmp;
	if (beta <= 1e+15) {
		tmp = (1.0 / (((0.5 * ((t_1 - (-2.0 * ((beta + 2.0) * (beta + (beta + 2.0))))) / t_1)) - 0.5) - (alpha / t_0))) / 2.0;
	} else {
		tmp = (1.0 / (0.5 + ((0.5 * alpha) / beta))) / 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 - (beta + 2.0d0)
    t_1 = t_0 ** 2.0d0
    if (beta <= 1d+15) then
        tmp = (1.0d0 / (((0.5d0 * ((t_1 - ((-2.0d0) * ((beta + 2.0d0) * (beta + (beta + 2.0d0))))) / t_1)) - 0.5d0) - (alpha / t_0))) / 2.0d0
    else
        tmp = (1.0d0 / (0.5d0 + ((0.5d0 * alpha) / beta))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = -beta - (beta + 2.0);
	double t_1 = Math.pow(t_0, 2.0);
	double tmp;
	if (beta <= 1e+15) {
		tmp = (1.0 / (((0.5 * ((t_1 - (-2.0 * ((beta + 2.0) * (beta + (beta + 2.0))))) / t_1)) - 0.5) - (alpha / t_0))) / 2.0;
	} else {
		tmp = (1.0 / (0.5 + ((0.5 * alpha) / beta))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = -beta - (beta + 2.0)
	t_1 = math.pow(t_0, 2.0)
	tmp = 0
	if beta <= 1e+15:
		tmp = (1.0 / (((0.5 * ((t_1 - (-2.0 * ((beta + 2.0) * (beta + (beta + 2.0))))) / t_1)) - 0.5) - (alpha / t_0))) / 2.0
	else:
		tmp = (1.0 / (0.5 + ((0.5 * alpha) / beta))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(-beta) - Float64(beta + 2.0))
	t_1 = t_0 ^ 2.0
	tmp = 0.0
	if (beta <= 1e+15)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(0.5 * Float64(Float64(t_1 - Float64(-2.0 * Float64(Float64(beta + 2.0) * Float64(beta + Float64(beta + 2.0))))) / t_1)) - 0.5) - Float64(alpha / t_0))) / 2.0);
	else
		tmp = Float64(Float64(1.0 / Float64(0.5 + Float64(Float64(0.5 * alpha) / beta))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = -beta - (beta + 2.0);
	t_1 = t_0 ^ 2.0;
	tmp = 0.0;
	if (beta <= 1e+15)
		tmp = (1.0 / (((0.5 * ((t_1 - (-2.0 * ((beta + 2.0) * (beta + (beta + 2.0))))) / t_1)) - 0.5) - (alpha / t_0))) / 2.0;
	else
		tmp = (1.0 / (0.5 + ((0.5 * alpha) / beta))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[((-beta) - N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[beta, 1e+15], N[(N[(1.0 / N[(N[(N[(0.5 * N[(N[(t$95$1 - N[(-2.0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 / N[(0.5 + N[(N[(0.5 * alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e15

   1. Initial program 70.6%

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

      1. +-commutative 70.6%

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

   3. Simplified 70.6%

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

      1. flip-+ 70.5%

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

      2. clear-num 70.5%

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

      3. sub-neg 70.5%

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

      4. associate-+l+ 70.5%

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

      5. metadata-eval 70.5%

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

      6. metadata-eval 70.5%

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

      7. sub-neg 70.5%

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

      8. pow2 70.5%

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

      9. associate-+l+ 70.5%

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

      10. metadata-eval 70.5%

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

   5. Applied egg-rr 70.5%

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

   6. Taylor expanded in alpha around -inf 99.9%

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

   if 1e15 < beta

   1. Initial program 85.1%

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

      1. +-commutative 85.1%

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

   3. Simplified 85.1%

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

      1. flip-+ 3.6%

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

      2. clear-num 3.6%

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

      3. sub-neg 3.6%

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

      4. associate-+l+ 3.6%

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

      5. metadata-eval 3.6%

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

      6. metadata-eval 3.6%

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

      7. sub-neg 3.6%

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

      8. pow2 3.6%

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

      9. associate-+l+ 3.6%

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

      10. metadata-eval 3.6%

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

   5. Applied egg-rr 3.6%

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

   6. Taylor expanded in alpha around -inf 50.5%

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

   7. Taylor expanded in beta around inf 99.9%

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

   8. Taylor expanded in beta around inf 99.9%

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

      1. associate-*r/ 99.9%

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

   10. Simplified 99.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{0.5 + \frac{0.5 \cdot \alpha}{\beta}}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{\left(0.5 \cdot \frac{{\left(\left(-\beta\right) - \left(\beta + 2\right)\right)}^{2} - -2 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\beta + 2\right)\right)\right)}{{\left(\left(-\beta\right) - \left(\beta + 2\right)\right)}^{2}} - 0.5\right) - \frac{\alpha}{\left(-\beta\right) - \left(\beta + 2\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{0.5 + \frac{0.5 \cdot \alpha}{\beta}}}{2}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ 2.0 alpha))))
   (if (<= (/ (- beta alpha) (+ 2.0 (+ beta alpha))) -0.99999)
     (/
      (+
       (* (/ (- (- -2.0 beta) beta) alpha) (/ (+ beta 2.0) alpha))
       (/ (+ beta (- beta -2.0)) alpha))
      2.0)
     (/ (+ (/ beta t_0) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (2.0 + alpha);
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.99999) {
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_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 + (2.0d0 + alpha)
    if (((beta - alpha) / (2.0d0 + (beta + alpha))) <= (-0.99999d0)) then
        tmp = ((((((-2.0d0) - beta) - beta) / alpha) * ((beta + 2.0d0) / alpha)) + ((beta + (beta - (-2.0d0))) / alpha)) / 2.0d0
    else
        tmp = ((beta / t_0) + (1.0d0 - (alpha / t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (2.0 + alpha);
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.99999) {
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (2.0 + alpha)
	tmp = 0
	if ((beta - alpha) / (2.0 + (beta + alpha))) <= -0.99999:
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0
	else:
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(2.0 + alpha))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))) <= -0.99999)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-2.0 - beta) - beta) / alpha) * Float64(Float64(beta + 2.0) / alpha)) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (2.0 + alpha);
	tmp = 0.0;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.99999)
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	else
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999990000000000046

   1. Initial program 9.3%

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

      1. +-commutative 9.3%

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

   3. Simplified 9.3%

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

   4. Taylor expanded in alpha around -inf 91.1%

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

   6. Simplified 100.0%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   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. Step-by-step derivation

      1. div-sub 99.9%

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

      2. associate-+l- 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

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

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(2 + \alpha\right)\\ t_1 := \frac{\beta}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.99999:\\ \;\;\;\;\frac{t_1 + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ 2.0 alpha))) (t_1 (/ beta t_0)))
   (if (<= (/ (- beta alpha) (+ 2.0 (+ beta alpha))) -0.99999)
     (/ (+ t_1 (/ (- beta -2.0) alpha)) 2.0)
     (/ (+ t_1 (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (2.0 + alpha);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.99999) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_1 + (1.0 - (alpha / t_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 + (2.0d0 + alpha)
    t_1 = beta / t_0
    if (((beta - alpha) / (2.0d0 + (beta + alpha))) <= (-0.99999d0)) then
        tmp = (t_1 + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = (t_1 + (1.0d0 - (alpha / t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (2.0 + alpha);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.99999) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (2.0 + alpha)
	t_1 = beta / t_0
	tmp = 0
	if ((beta - alpha) / (2.0 + (beta + alpha))) <= -0.99999:
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(2.0 + alpha))
	t_1 = Float64(beta / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))) <= -0.99999)
		tmp = Float64(Float64(t_1 + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(t_1 + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (2.0 + alpha);
	t_1 = beta / t_0;
	tmp = 0.0;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.99999)
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(t$95$1 + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$1 + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999990000000000046

   1. Initial program 9.3%

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

      1. +-commutative 9.3%

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

   3. Simplified 9.3%

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

      1. div-sub 9.3%

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

      2. associate-+l- 12.4%

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

      3. associate-+l+ 12.4%

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

      4. associate-+l+ 12.4%

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

   5. Applied egg-rr 12.4%

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

   6. Taylor expanded in alpha around inf 97.8%

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

      1. associate-*r/ 97.8%

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

      2. distribute-lft-in 97.8%

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

      3. metadata-eval 97.8%

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

      4. mul-1-neg 97.8%

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

      5. sub-neg 97.8%

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

   8. Simplified 97.8%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   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. Step-by-step derivation

      1. div-sub 99.9%

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

      2. associate-+l- 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.3%

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

Alternative 4: 99.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.99999)
     (/ (+ (/ beta (+ beta (+ 2.0 alpha))) (/ (- beta -2.0) alpha)) 2.0)
     (/ (+ 1.0 t_0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.99999) {
		tmp = ((beta / (beta + (2.0 + alpha))) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + t_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) / (2.0d0 + (beta + alpha))
    if (t_0 <= (-0.99999d0)) then
        tmp = ((beta / (beta + (2.0d0 + alpha))) + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = (1.0d0 + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.99999) {
		tmp = ((beta / (beta + (2.0 + alpha))) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / (2.0 + (beta + alpha))
	tmp = 0
	if t_0 <= -0.99999:
		tmp = ((beta / (beta + (2.0 + alpha))) + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = (1.0 + t_0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.99999)
		tmp = Float64(Float64(Float64(beta / Float64(beta + Float64(2.0 + alpha))) + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	tmp = 0.0;
	if (t_0 <= -0.99999)
		tmp = ((beta / (beta + (2.0 + alpha))) + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = (1.0 + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99999], N[(N[(N[(beta / N[(beta + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999990000000000046

   1. Initial program 9.3%

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

      1. +-commutative 9.3%

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

   3. Simplified 9.3%

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

      1. div-sub 9.3%

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

      2. associate-+l- 12.4%

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

      3. associate-+l+ 12.4%

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

      4. associate-+l+ 12.4%

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

   5. Applied egg-rr 12.4%

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

   6. Taylor expanded in alpha around inf 97.8%

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

      1. associate-*r/ 97.8%

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

      2. distribute-lft-in 97.8%

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

      3. metadata-eval 97.8%

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

      4. mul-1-neg 97.8%

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

      5. sub-neg 97.8%

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

   8. Simplified 97.8%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

4. Final simplification 99.3%

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

Alternative 5: 99.6% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.99999)
     (+ (/ beta alpha) (/ 1.0 alpha))
     (/ (+ 1.0 t_0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.99999) {
		tmp = (beta / alpha) + (1.0 / alpha);
	} else {
		tmp = (1.0 + t_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) / (2.0d0 + (beta + alpha))
    if (t_0 <= (-0.99999d0)) then
        tmp = (beta / alpha) + (1.0d0 / alpha)
    else
        tmp = (1.0d0 + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.99999) {
		tmp = (beta / alpha) + (1.0 / alpha);
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / (2.0 + (beta + alpha))
	tmp = 0
	if t_0 <= -0.99999:
		tmp = (beta / alpha) + (1.0 / alpha)
	else:
		tmp = (1.0 + t_0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.99999)
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	else
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	tmp = 0.0;
	if (t_0 <= -0.99999)
		tmp = (beta / alpha) + (1.0 / alpha);
	else
		tmp = (1.0 + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99999], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999990000000000046

   1. Initial program 9.3%

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

      1. +-commutative 9.3%

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

   3. Simplified 9.3%

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

   4. Taylor expanded in alpha around -inf 97.8%

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

      1. associate-*r/ 97.8%

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

      2. sub-neg 97.8%

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

      3. mul-1-neg 97.8%

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

      4. distribute-lft-in 97.8%

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

      5. neg-mul-1 97.8%

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

      6. mul-1-neg 97.8%

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

      7. remove-double-neg 97.8%

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

      8. neg-mul-1 97.8%

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

      9. mul-1-neg 97.8%

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

      10. remove-double-neg 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in beta around 0 97.8%

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

      1. +-commutative 97.8%

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

   9. Simplified 97.8%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

4. Final simplification 99.3%

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

Alternative 6: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 370000000:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{0.5 - \frac{\alpha}{\left(-\beta\right) - \left(\beta + 2\right)}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 370000000.0)
   (/ (+ 1.0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))) 2.0)
   (/ (/ 1.0 (- 0.5 (/ alpha (- (- beta) (+ beta 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 370000000.0) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + (beta + alpha)))) / 2.0;
	} else {
		tmp = (1.0 / (0.5 - (alpha / (-beta - (beta + 2.0))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 370000000.0d0) then
        tmp = (1.0d0 + ((beta - alpha) / (2.0d0 + (beta + alpha)))) / 2.0d0
    else
        tmp = (1.0d0 / (0.5d0 - (alpha / (-beta - (beta + 2.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 370000000.0) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + (beta + alpha)))) / 2.0;
	} else {
		tmp = (1.0 / (0.5 - (alpha / (-beta - (beta + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 370000000.0:
		tmp = (1.0 + ((beta - alpha) / (2.0 + (beta + alpha)))) / 2.0
	else:
		tmp = (1.0 / (0.5 - (alpha / (-beta - (beta + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 370000000.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 / Float64(0.5 - Float64(alpha / Float64(Float64(-beta) - Float64(beta + 2.0))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 370000000.0)
		tmp = (1.0 + ((beta - alpha) / (2.0 + (beta + alpha)))) / 2.0;
	else
		tmp = (1.0 / (0.5 - (alpha / (-beta - (beta + 2.0))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 370000000.0], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 / N[(0.5 - N[(alpha / N[((-beta) - N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.7e8

   1. Initial program 99.9%

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

   if 3.7e8 < alpha

   1. Initial program 25.6%

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

      1. +-commutative 25.6%

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

   3. Simplified 25.6%

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

      1. flip-+ 9.7%

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

      2. clear-num 9.7%

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

      3. sub-neg 9.7%

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

      4. associate-+l+ 9.7%

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

      5. metadata-eval 9.7%

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

      6. metadata-eval 9.7%

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

      7. sub-neg 9.7%

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

      8. pow2 9.7%

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

      9. associate-+l+ 9.7%

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

      10. metadata-eval 9.7%

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

   5. Applied egg-rr 9.7%

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

   6. Taylor expanded in alpha around -inf 78.3%

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

   7. Taylor expanded in beta around inf 98.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 370000000:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{0.5 - \frac{\alpha}{\left(-\beta\right) - \left(\beta + 2\right)}}}{2}\\ \end{array} \]

Alternative 7: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq -2 \cdot 10^{-209}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -1 \cdot 10^{-291}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha -2e-209)
   0.5
   (if (<= alpha -1e-291)
     1.0
     (if (<= alpha 50000.0) 0.5 (+ (/ beta alpha) (/ 1.0 alpha))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -2e-209) {
		tmp = 0.5;
	} else if (alpha <= -1e-291) {
		tmp = 1.0;
	} else if (alpha <= 50000.0) {
		tmp = 0.5;
	} else {
		tmp = (beta / alpha) + (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-209)) then
        tmp = 0.5d0
    else if (alpha <= (-1d-291)) then
        tmp = 1.0d0
    else if (alpha <= 50000.0d0) then
        tmp = 0.5d0
    else
        tmp = (beta / alpha) + (1.0d0 / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -2e-209) {
		tmp = 0.5;
	} else if (alpha <= -1e-291) {
		tmp = 1.0;
	} else if (alpha <= 50000.0) {
		tmp = 0.5;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= -2e-209:
		tmp = 0.5
	elif alpha <= -1e-291:
		tmp = 1.0
	elif alpha <= 50000.0:
		tmp = 0.5
	else:
		tmp = (beta / alpha) + (1.0 / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= -2e-209)
		tmp = 0.5;
	elseif (alpha <= -1e-291)
		tmp = 1.0;
	elseif (alpha <= 50000.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= -2e-209)
		tmp = 0.5;
	elseif (alpha <= -1e-291)
		tmp = 1.0;
	elseif (alpha <= 50000.0)
		tmp = 0.5;
	else
		tmp = (beta / alpha) + (1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, -2e-209], 0.5, If[LessEqual[alpha, -1e-291], 1.0, If[LessEqual[alpha, 50000.0], 0.5, N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -2.0000000000000001e-209 or -9.99999999999999962e-292 < alpha < 5e4

   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. Taylor expanded in beta around 0 72.7%

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

      1. +-commutative 72.7%

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

   6. Simplified 72.7%

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

   7. Taylor expanded in alpha around 0 69.9%

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

   if -2.0000000000000001e-209 < alpha < -9.99999999999999962e-292

   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. Taylor expanded in beta around inf 67.6%

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

   if 5e4 < alpha

   1. Initial program 26.2%

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

      1. +-commutative 26.2%

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

   3. Simplified 26.2%

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

   4. Taylor expanded in alpha around -inf 81.1%

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

      1. associate-*r/ 81.1%

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

      2. sub-neg 81.1%

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

      3. mul-1-neg 81.1%

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

      4. distribute-lft-in 81.1%

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

      5. neg-mul-1 81.1%

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

      6. mul-1-neg 81.1%

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

      7. remove-double-neg 81.1%

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

      8. neg-mul-1 81.1%

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

      9. mul-1-neg 81.1%

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

      10. remove-double-neg 81.1%

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

   6. Simplified 81.1%

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

   7. Taylor expanded in beta around 0 81.1%

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

      1. +-commutative 81.1%

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

   9. Simplified 81.1%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -2 \cdot 10^{-209}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -1 \cdot 10^{-291}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]

Alternative 8: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq 1.2 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)))
   (if (<= beta 1.2e-219)
     t_0
     (if (<= beta 6.2e-145) (/ 1.0 alpha) (if (<= beta 2.0) t_0 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= 1.2e-219) {
		tmp = t_0;
	} else if (beta <= 6.2e-145) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    if (beta <= 1.2d-219) then
        tmp = t_0
    else if (beta <= 6.2d-145) then
        tmp = 1.0d0 / alpha
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= 1.2e-219) {
		tmp = t_0;
	} else if (beta <= 6.2e-145) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (beta * 0.5)) / 2.0
	tmp = 0
	if beta <= 1.2e-219:
		tmp = t_0
	elif beta <= 6.2e-145:
		tmp = 1.0 / alpha
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0)
	tmp = 0.0
	if (beta <= 1.2e-219)
		tmp = t_0;
	elseif (beta <= 6.2e-145)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta * 0.5)) / 2.0;
	tmp = 0.0;
	if (beta <= 1.2e-219)
		tmp = t_0;
	elseif (beta <= 6.2e-145)
		tmp = 1.0 / alpha;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.2e-219], t$95$0, If[LessEqual[beta, 6.2e-145], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 2.0], t$95$0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.20000000000000007e-219 or 6.20000000000000001e-145 < beta < 2

   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. Taylor expanded in alpha around 0 69.1%

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

   5. Taylor expanded in beta around 0 68.4%

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

   if 1.20000000000000007e-219 < beta < 6.20000000000000001e-145

   1. Initial program 39.8%

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

      1. +-commutative 39.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in alpha around -inf 65.9%

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

      1. associate-*r/ 65.9%

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

      2. sub-neg 65.9%

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

      3. mul-1-neg 65.9%

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

      4. distribute-lft-in 65.9%

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

      5. neg-mul-1 65.9%

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

      6. mul-1-neg 65.9%

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

      7. remove-double-neg 65.9%

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

      8. neg-mul-1 65.9%

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

      9. mul-1-neg 65.9%

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

      10. remove-double-neg 65.9%

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

   6. Simplified 65.9%

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

   7. Taylor expanded in beta around 0 65.9%

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

   if 2 < beta

   1. Initial program 84.6%

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

      1. +-commutative 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in beta around inf 82.2%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{-219}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - 0.5 \cdot \alpha}{2}\\ \mathbf{if}\;\alpha \leq -2.05 \cdot 10^{-215}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq -1.9 \cdot 10^{-291}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.95:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (* 0.5 alpha)) 2.0)))
   (if (<= alpha -2.05e-215)
     t_0
     (if (<= alpha -1.9e-291)
       1.0
       (if (<= alpha 1.95) t_0 (+ (/ beta alpha) (/ 1.0 alpha)))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 - (0.5 * alpha)) / 2.0;
	double tmp;
	if (alpha <= -2.05e-215) {
		tmp = t_0;
	} else if (alpha <= -1.9e-291) {
		tmp = 1.0;
	} else if (alpha <= 1.95) {
		tmp = t_0;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	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 = (1.0d0 - (0.5d0 * alpha)) / 2.0d0
    if (alpha <= (-2.05d-215)) then
        tmp = t_0
    else if (alpha <= (-1.9d-291)) then
        tmp = 1.0d0
    else if (alpha <= 1.95d0) then
        tmp = t_0
    else
        tmp = (beta / alpha) + (1.0d0 / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 - (0.5 * alpha)) / 2.0;
	double tmp;
	if (alpha <= -2.05e-215) {
		tmp = t_0;
	} else if (alpha <= -1.9e-291) {
		tmp = 1.0;
	} else if (alpha <= 1.95) {
		tmp = t_0;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 - (0.5 * alpha)) / 2.0
	tmp = 0
	if alpha <= -2.05e-215:
		tmp = t_0
	elif alpha <= -1.9e-291:
		tmp = 1.0
	elif alpha <= 1.95:
		tmp = t_0
	else:
		tmp = (beta / alpha) + (1.0 / alpha)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 - Float64(0.5 * alpha)) / 2.0)
	tmp = 0.0
	if (alpha <= -2.05e-215)
		tmp = t_0;
	elseif (alpha <= -1.9e-291)
		tmp = 1.0;
	elseif (alpha <= 1.95)
		tmp = t_0;
	else
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 - (0.5 * alpha)) / 2.0;
	tmp = 0.0;
	if (alpha <= -2.05e-215)
		tmp = t_0;
	elseif (alpha <= -1.9e-291)
		tmp = 1.0;
	elseif (alpha <= 1.95)
		tmp = t_0;
	else
		tmp = (beta / alpha) + (1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 - N[(0.5 * alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, -2.05e-215], t$95$0, If[LessEqual[alpha, -1.9e-291], 1.0, If[LessEqual[alpha, 1.95], t$95$0, N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -2.04999999999999992e-215 or -1.8999999999999999e-291 < alpha < 1.94999999999999996

   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. Taylor expanded in beta around 0 73.4%

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

      1. +-commutative 73.4%

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

   6. Simplified 73.4%

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

   7. Taylor expanded in alpha around 0 71.9%

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

   if -2.04999999999999992e-215 < alpha < -1.8999999999999999e-291

   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. Taylor expanded in beta around inf 67.6%

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

   if 1.94999999999999996 < alpha

   1. Initial program 28.7%

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

      1. +-commutative 28.7%

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

   3. Simplified 28.7%

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

   4. Taylor expanded in alpha around -inf 78.7%

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

      1. associate-*r/ 78.7%

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

      2. sub-neg 78.7%

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

      3. mul-1-neg 78.7%

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

      4. distribute-lft-in 78.7%

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

      5. neg-mul-1 78.7%

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

      6. mul-1-neg 78.7%

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

      7. remove-double-neg 78.7%

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

      8. neg-mul-1 78.7%

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

      9. mul-1-neg 78.7%

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

      10. remove-double-neg 78.7%

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

   6. Simplified 78.7%

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

   7. Taylor expanded in beta around 0 78.7%

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

      1. +-commutative 78.7%

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

   9. Simplified 78.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -2.05 \cdot 10^{-215}:\\ \;\;\;\;\frac{1 - 0.5 \cdot \alpha}{2}\\ \mathbf{elif}\;\alpha \leq -1.9 \cdot 10^{-291}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.95:\\ \;\;\;\;\frac{1 - 0.5 \cdot \alpha}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]

Alternative 10: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.56:\\ \;\;\;\;\frac{\frac{1}{1 + 0.5 \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{0.5 + \frac{0.5 \cdot \alpha}{\beta}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.56)
   (/ (/ 1.0 (+ 1.0 (* 0.5 alpha))) 2.0)
   (/ (/ 1.0 (+ 0.5 (/ (* 0.5 alpha) beta))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.56) {
		tmp = (1.0 / (1.0 + (0.5 * alpha))) / 2.0;
	} else {
		tmp = (1.0 / (0.5 + ((0.5 * alpha) / beta))) / 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 <= 1.56d0) then
        tmp = (1.0d0 / (1.0d0 + (0.5d0 * alpha))) / 2.0d0
    else
        tmp = (1.0d0 / (0.5d0 + ((0.5d0 * alpha) / beta))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.56) {
		tmp = (1.0 / (1.0 + (0.5 * alpha))) / 2.0;
	} else {
		tmp = (1.0 / (0.5 + ((0.5 * alpha) / beta))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.56:
		tmp = (1.0 / (1.0 + (0.5 * alpha))) / 2.0
	else:
		tmp = (1.0 / (0.5 + ((0.5 * alpha) / beta))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.56)
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(0.5 * alpha))) / 2.0);
	else
		tmp = Float64(Float64(1.0 / Float64(0.5 + Float64(Float64(0.5 * alpha) / beta))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.56)
		tmp = (1.0 / (1.0 + (0.5 * alpha))) / 2.0;
	else
		tmp = (1.0 / (0.5 + ((0.5 * alpha) / beta))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.56], N[(N[(1.0 / N[(1.0 + N[(0.5 * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 / N[(0.5 + N[(N[(0.5 * alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5600000000000001

   1. Initial program 70.6%

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

      1. +-commutative 70.6%

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

   3. Simplified 70.6%

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

      1. flip-+ 70.6%

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

      2. clear-num 70.6%

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

      3. sub-neg 70.6%

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

      4. associate-+l+ 70.6%

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

      5. metadata-eval 70.6%

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

      6. metadata-eval 70.6%

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

      7. sub-neg 70.6%

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

      8. pow2 70.6%

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

      9. associate-+l+ 70.6%

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

      10. metadata-eval 70.6%

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

   5. Applied egg-rr 70.6%

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

   6. Taylor expanded in alpha around -inf 99.9%

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

   7. Taylor expanded in beta around 0 98.0%

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

   if 1.5600000000000001 < beta

   1. Initial program 84.6%

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

      1. +-commutative 84.6%

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

   3. Simplified 84.6%

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

      1. flip-+ 5.6%

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

      2. clear-num 5.6%

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

      3. sub-neg 5.6%

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

      4. associate-+l+ 5.6%

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

      5. metadata-eval 5.6%

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

      6. metadata-eval 5.6%

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

      7. sub-neg 5.6%

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

      8. pow2 5.6%

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

      9. associate-+l+ 5.6%

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

      10. metadata-eval 5.6%

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

   5. Applied egg-rr 5.6%

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

   6. Taylor expanded in alpha around -inf 52.0%

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

   7. Taylor expanded in beta around inf 99.1%

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

   8. Taylor expanded in beta around inf 98.4%

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

      1. associate-*r/ 98.4%

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

   10. Simplified 98.4%

       \[\leadsto \frac{\frac{1}{\color{blue}{0.5 + \frac{0.5 \cdot \alpha}{\beta}}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.56:\\ \;\;\;\;\frac{\frac{1}{1 + 0.5 \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{0.5 + \frac{0.5 \cdot \alpha}{\beta}}}{2}\\ \end{array} \]

Alternative 11: 93.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 114000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 114000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (+ (/ beta alpha) (/ 1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 114000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + (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 <= 114000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta / alpha) + (1.0d0 / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 114000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 114000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta / alpha) + (1.0 / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 114000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 114000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta / alpha) + (1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 114000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 114000

   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. Taylor expanded in alpha around 0 97.4%

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

   if 114000 < alpha

   1. Initial program 26.2%

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

      1. +-commutative 26.2%

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

   3. Simplified 26.2%

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

   4. Taylor expanded in alpha around -inf 81.1%

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

      1. associate-*r/ 81.1%

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

      2. sub-neg 81.1%

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

      3. mul-1-neg 81.1%

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

      4. distribute-lft-in 81.1%

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

      5. neg-mul-1 81.1%

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

      6. mul-1-neg 81.1%

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

      7. remove-double-neg 81.1%

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

      8. neg-mul-1 81.1%

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

      9. mul-1-neg 81.1%

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

      10. remove-double-neg 81.1%

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

   6. Simplified 81.1%

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

   7. Taylor expanded in beta around 0 81.1%

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

      1. +-commutative 81.1%

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

   9. Simplified 81.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 114000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]

Alternative 12: 93.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{1}{1 + 0.5 \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.1e-16)
   (/ (/ 1.0 (+ 1.0 (* 0.5 alpha))) 2.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.1e-16) {
		tmp = (1.0 / (1.0 + (0.5 * alpha))) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.1d-16) then
        tmp = (1.0d0 / (1.0d0 + (0.5d0 * alpha))) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.1e-16) {
		tmp = (1.0 / (1.0 + (0.5 * alpha))) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.1e-16:
		tmp = (1.0 / (1.0 + (0.5 * alpha))) / 2.0
	else:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.1e-16)
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(0.5 * alpha))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.1e-16)
		tmp = (1.0 / (1.0 + (0.5 * alpha))) / 2.0;
	else
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.1e-16], N[(N[(1.0 / N[(1.0 + N[(0.5 * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1e-16

   1. Initial program 69.3%

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

      1. +-commutative 69.3%

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

   3. Simplified 69.3%

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

      1. flip-+ 69.3%

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

      2. clear-num 69.2%

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

      3. sub-neg 69.2%

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

      4. associate-+l+ 69.2%

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

      5. metadata-eval 69.2%

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

      6. metadata-eval 69.2%

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

      7. sub-neg 69.2%

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

      8. pow2 69.2%

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

      9. associate-+l+ 69.2%

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

      10. metadata-eval 69.2%

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

   5. Applied egg-rr 69.2%

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

   6. Taylor expanded in alpha around -inf 99.9%

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

   7. Taylor expanded in beta around 0 99.5%

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

   if 1.1e-16 < beta

   1. Initial program 85.7%

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

      1. +-commutative 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in alpha around 0 84.1%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{1}{1 + 0.5 \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \]

Alternative 13: 68.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7 \cdot 10^{-222}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7e-222)
   0.5
   (if (<= beta 9.2e-145) (/ 1.0 alpha) (if (<= beta 2.0) 0.5 1.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7e-222) {
		tmp = 0.5;
	} else if (beta <= 9.2e-145) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.7d-222) then
        tmp = 0.5d0
    else if (beta <= 9.2d-145) then
        tmp = 1.0d0 / alpha
    else if (beta <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7e-222) {
		tmp = 0.5;
	} else if (beta <= 9.2e-145) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.7e-222:
		tmp = 0.5
	elif beta <= 9.2e-145:
		tmp = 1.0 / alpha
	elif beta <= 2.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7e-222)
		tmp = 0.5;
	elseif (beta <= 9.2e-145)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7e-222)
		tmp = 0.5;
	elseif (beta <= 9.2e-145)
		tmp = 1.0 / alpha;
	elseif (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.7e-222], 0.5, If[LessEqual[beta, 9.2e-145], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 2.0], 0.5, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.7000000000000001e-222 or 9.20000000000000028e-145 < beta < 2

   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. Taylor expanded in beta around 0 70.7%

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

      1. +-commutative 70.7%

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

   6. Simplified 70.7%

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

   7. Taylor expanded in alpha around 0 67.1%

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

   if 1.7000000000000001e-222 < beta < 9.20000000000000028e-145

   1. Initial program 39.8%

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

      1. +-commutative 39.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in alpha around -inf 65.9%

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

      1. associate-*r/ 65.9%

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

      2. sub-neg 65.9%

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

      3. mul-1-neg 65.9%

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

      4. distribute-lft-in 65.9%

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

      5. neg-mul-1 65.9%

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

      6. mul-1-neg 65.9%

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

      7. remove-double-neg 65.9%

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

      8. neg-mul-1 65.9%

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

      9. mul-1-neg 65.9%

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

      10. remove-double-neg 65.9%

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

   6. Simplified 65.9%

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

   7. Taylor expanded in beta around 0 65.9%

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

   if 2 < beta

   1. Initial program 84.6%

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

      1. +-commutative 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in beta around inf 82.2%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.7 \cdot 10^{-222}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 68.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 50000.0) 0.5 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 50000.0) {
		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 <= 50000.0d0) 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 <= 50000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 50000.0:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 50000.0)
		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 <= 50000.0)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 50000.0], 0.5, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5e4

   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. Taylor expanded in beta around 0 68.2%

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

      1. +-commutative 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in alpha around 0 65.8%

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

   if 5e4 < alpha

   1. Initial program 26.2%

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

      1. +-commutative 26.2%

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

   3. Simplified 26.2%

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

   4. Taylor expanded in alpha around -inf 81.1%

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

      1. associate-*r/ 81.1%

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

      2. sub-neg 81.1%

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

      3. mul-1-neg 81.1%

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

      4. distribute-lft-in 81.1%

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

      5. neg-mul-1 81.1%

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

      6. mul-1-neg 81.1%

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

      7. remove-double-neg 81.1%

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

      8. neg-mul-1 81.1%

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

      9. mul-1-neg 81.1%

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

      10. remove-double-neg 81.1%

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

   6. Simplified 81.1%

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

   7. Taylor expanded in beta around 0 63.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 15: 49.9% 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 75.8%

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

   1. +-commutative 75.8%

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

3. Simplified 75.8%

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

4. Taylor expanded in beta around 0 48.5%

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

   1. +-commutative 48.5%

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

6. Simplified 48.5%

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

7. Taylor expanded in alpha around 0 46.9%

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

8. Final simplification 46.9%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023274 
(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))