Octave 3.8, jcobi/1

Percentage Accurate: 74.3% → 99.9%

Time: 10.7s

Alternatives: 13

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: 74.3% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\beta\right) - \left(\beta + 2\right)\\ t_1 := \frac{\alpha}{t_0}\\ t_2 := {t_0}^{2}\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{\left(0.5 \cdot \frac{t_2 - -2 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\beta + 2\right)\right)\right)}{t_2} - 0.5\right) - t_1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{0.5 - t_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (- beta) (+ beta 2.0)))
        (t_1 (/ alpha t_0))
        (t_2 (pow t_0 2.0)))
   (if (<= beta 5e+15)
     (/
      (/
       1.0
       (-
        (-
         (*
          0.5
          (/ (- t_2 (* -2.0 (* (+ beta 2.0) (+ beta (+ beta 2.0))))) t_2))
         0.5)
        t_1))
      2.0)
     (/ (/ 1.0 (- 0.5 t_1)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = -beta - (beta + 2.0);
	double t_1 = alpha / t_0;
	double t_2 = pow(t_0, 2.0);
	double tmp;
	if (beta <= 5e+15) {
		tmp = (1.0 / (((0.5 * ((t_2 - (-2.0 * ((beta + 2.0) * (beta + (beta + 2.0))))) / t_2)) - 0.5) - t_1)) / 2.0;
	} else {
		tmp = (1.0 / (0.5 - t_1)) / 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) :: t_2
    real(8) :: tmp
    t_0 = -beta - (beta + 2.0d0)
    t_1 = alpha / t_0
    t_2 = t_0 ** 2.0d0
    if (beta <= 5d+15) then
        tmp = (1.0d0 / (((0.5d0 * ((t_2 - ((-2.0d0) * ((beta + 2.0d0) * (beta + (beta + 2.0d0))))) / t_2)) - 0.5d0) - t_1)) / 2.0d0
    else
        tmp = (1.0d0 / (0.5d0 - t_1)) / 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 = alpha / t_0;
	double t_2 = Math.pow(t_0, 2.0);
	double tmp;
	if (beta <= 5e+15) {
		tmp = (1.0 / (((0.5 * ((t_2 - (-2.0 * ((beta + 2.0) * (beta + (beta + 2.0))))) / t_2)) - 0.5) - t_1)) / 2.0;
	} else {
		tmp = (1.0 / (0.5 - t_1)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = -beta - (beta + 2.0)
	t_1 = alpha / t_0
	t_2 = math.pow(t_0, 2.0)
	tmp = 0
	if beta <= 5e+15:
		tmp = (1.0 / (((0.5 * ((t_2 - (-2.0 * ((beta + 2.0) * (beta + (beta + 2.0))))) / t_2)) - 0.5) - t_1)) / 2.0
	else:
		tmp = (1.0 / (0.5 - t_1)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(-beta) - Float64(beta + 2.0))
	t_1 = Float64(alpha / t_0)
	t_2 = t_0 ^ 2.0
	tmp = 0.0
	if (beta <= 5e+15)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(0.5 * Float64(Float64(t_2 - Float64(-2.0 * Float64(Float64(beta + 2.0) * Float64(beta + Float64(beta + 2.0))))) / t_2)) - 0.5) - t_1)) / 2.0);
	else
		tmp = Float64(Float64(1.0 / Float64(0.5 - t_1)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = -beta - (beta + 2.0);
	t_1 = alpha / t_0;
	t_2 = t_0 ^ 2.0;
	tmp = 0.0;
	if (beta <= 5e+15)
		tmp = (1.0 / (((0.5 * ((t_2 - (-2.0 * ((beta + 2.0) * (beta + (beta + 2.0))))) / t_2)) - 0.5) - t_1)) / 2.0;
	else
		tmp = (1.0 / (0.5 - t_1)) / 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[(alpha / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[beta, 5e+15], N[(N[(1.0 / N[(N[(N[(0.5 * N[(N[(t$95$2 - N[(-2.0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 / N[(0.5 - t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e15

   1. Initial program 67.5%

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

      1. +-commutative 67.5%

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

   3. Simplified 67.5%

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

      1. flip-+ 67.1%

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

      2. clear-num 67.2%

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

      3. sub-neg 67.2%

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

      4. associate-+l+ 67.2%

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

      5. metadata-eval 67.2%

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

      6. metadata-eval 67.2%

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

      7. sub-neg 67.2%

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

      8. pow2 67.2%

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

      9. associate-+l+ 67.2%

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

      10. metadata-eval 67.2%

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

   5. Applied egg-rr 67.2%

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

   6. Taylor expanded in alpha around -inf 99.9%

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

   if 5e15 < beta

   1. Initial program 88.2%

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

      1. +-commutative 88.2%

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

   3. Simplified 88.2%

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

      1. flip-+ 6.0%

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

      2. clear-num 6.0%

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

      3. sub-neg 6.0%

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

      4. associate-+l+ 6.0%

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

      5. metadata-eval 6.0%

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

      6. metadata-eval 6.0%

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

      7. sub-neg 6.0%

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

      8. pow2 6.0%

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

      9. associate-+l+ 6.0%

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

      10. metadata-eval 6.0%

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

   5. Applied egg-rr 6.0%

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

   6. Taylor expanded in alpha around -inf 41.3%

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

   7. Taylor expanded in beta around inf 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{\left(0.5 \cdot \frac{{\left(\left(-\beta\right) - \left(\beta + 2\right)\right)}^{2} - -2 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\beta + 2\right)\right)\right)}{{\left(\left(-\beta\right) - \left(\beta + 2\right)\right)}^{2}} - 0.5\right) - \frac{\alpha}{\left(-\beta\right) - \left(\beta + 2\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{0.5 - \frac{\alpha}{\left(-\beta\right) - \left(\beta + 2\right)}}}{2}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t_0 \leq -0.9996:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.9996)
     (/
      (fma
       (/ (- (- -2.0 beta) beta) alpha)
       (/ (+ beta 2.0) alpha)
       (/ (+ beta (+ 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 <= -0.9996) {
		tmp = fma((((-2.0 - beta) - beta) / alpha), ((beta + 2.0) / alpha), ((beta + (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 <= -0.9996)
		tmp = Float64(fma(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(1.0 + t_0) / 2.0);
	end
	return 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, -0.9996], N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $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)) < -0.99960000000000004

   1. Initial program 10.0%

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

      1. +-commutative 10.0%

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

   3. Simplified 10.0%

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

   4. Taylor expanded in alpha around -inf 93.2%

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

   6. Simplified 99.8%

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

   if -0.99960000000000004 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.9996:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}}{2}\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(2 + \alpha\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{t_0}\right) - \frac{\alpha}{t_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ 2.0 alpha))))
   (if (<= (/ (- beta alpha) (+ 2.0 (+ beta alpha))) -0.99999999)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (- (+ 1.0 (/ beta t_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))) <= -0.99999999) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_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))) <= (-0.99999999d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((1.0d0 + (beta / t_0)) - (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))) <= -0.99999999) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_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))) <= -0.99999999:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = ((1.0 + (beta / t_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))) <= -0.99999999)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(beta / t_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))) <= -0.99999999)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = ((1.0 + (beta / t_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], -0.99999999], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999998999999995

   1. Initial program 7.8%

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

      1. +-commutative 7.8%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in alpha around inf 98.4%

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

   if -0.99999998999999995 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

   3. Simplified 99.3%

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

      1. +-commutative 99.3%

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

      2. div-sub 99.3%

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

      3. associate-+r- 99.3%

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

      4. associate-+l+ 99.3%

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

      5. associate-+l+ 99.3%

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

   5. Applied egg-rr 99.3%

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

4. Final simplification 99.1%

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

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(2 + \alpha\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ 2.0 alpha))))
   (if (<= (/ (- beta alpha) (+ 2.0 (+ beta alpha))) -0.99999999)
     (/ (/ (+ 2.0 (* 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))) <= -0.99999999) {
		tmp = ((2.0 + (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))) <= (-0.99999999d0)) then
        tmp = ((2.0d0 + (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))) <= -0.99999999) {
		tmp = ((2.0 + (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))) <= -0.99999999:
		tmp = ((2.0 + (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))) <= -0.99999999)
		tmp = Float64(Float64(Float64(2.0 + 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))) <= -0.99999999)
		tmp = ((2.0 + (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], -0.99999999], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $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)) < -0.99999998999999995

   1. Initial program 7.8%

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

      1. +-commutative 7.8%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in alpha around inf 98.4%

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

   if -0.99999998999999995 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

   3. Simplified 99.3%

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

      1. div-sub 99.3%

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

      2. associate-+l- 99.4%

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

      3. associate-+l+ 99.4%

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

      4. associate-+l+ 99.4%

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

   5. Applied egg-rr 99.4%

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

4. Final simplification 99.1%

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

Alternative 5: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t_0 \leq -0.99999999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.99999999)
     (/ (/ (+ 2.0 (* 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 <= -0.99999999) {
		tmp = ((2.0 + (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 <= (-0.99999999d0)) then
        tmp = ((2.0d0 + (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 <= -0.99999999) {
		tmp = ((2.0 + (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 <= -0.99999999:
		tmp = ((2.0 + (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 <= -0.99999999)
		tmp = Float64(Float64(Float64(2.0 + 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 <= -0.99999999)
		tmp = ((2.0 + (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, -0.99999999], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $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)) < -0.99999998999999995

   1. Initial program 7.8%

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

      1. +-commutative 7.8%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in alpha around inf 98.4%

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

   if -0.99999998999999995 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.3%

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

4. Final simplification 99.1%

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

Alternative 6: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 280000000:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{0.5 - \frac{\alpha}{\left(-\beta\right) - \left(\beta + 2\right)}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 280000000.0)
   (/ (+ 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 <= 280000000.0) {
		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 <= 280000000.0d0) 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 <= 280000000.0) {
		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 <= 280000000.0:
		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 <= 280000000.0)
		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 <= 280000000.0)
		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, 280000000.0], 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 < 2.8e8

   1. Initial program 99.7%

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

   if 2.8e8 < alpha

   1. Initial program 23.7%

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

      1. +-commutative 23.7%

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

   3. Simplified 23.7%

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

      1. flip-+ 10.5%

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

      2. clear-num 10.5%

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

      3. sub-neg 10.5%

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

      4. associate-+l+ 10.5%

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

      5. metadata-eval 10.5%

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

      6. metadata-eval 10.5%

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

      7. sub-neg 10.5%

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

      8. pow2 10.5%

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

      9. associate-+l+ 10.5%

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

      10. metadata-eval 10.5%

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

   5. Applied egg-rr 10.5%

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

   6. Taylor expanded in alpha around -inf 80.6%

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

   7. Taylor expanded in beta around inf 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 280000000:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{0.5 - \frac{\alpha}{\left(-\beta\right) - \left(\beta + 2\right)}}}{2}\\ \end{array} \]

Alternative 7: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - 0.5 \cdot \alpha}{2}\\ \mathbf{if}\;\alpha \leq -1.9 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq -5.8 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.9:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (* 0.5 alpha)) 2.0)))
   (if (<= alpha -1.9e-114)
     t_0
     (if (<= alpha -5.8e-170)
       1.0
       (if (<= alpha 0.9) t_0 (/ (/ 2.0 alpha) 2.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 - (0.5 * alpha)) / 2.0;
	double tmp;
	if (alpha <= -1.9e-114) {
		tmp = t_0;
	} else if (alpha <= -5.8e-170) {
		tmp = 1.0;
	} else if (alpha <= 0.9) {
		tmp = t_0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - (0.5d0 * alpha)) / 2.0d0
    if (alpha <= (-1.9d-114)) then
        tmp = t_0
    else if (alpha <= (-5.8d-170)) then
        tmp = 1.0d0
    else if (alpha <= 0.9d0) then
        tmp = t_0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    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 <= -1.9e-114) {
		tmp = t_0;
	} else if (alpha <= -5.8e-170) {
		tmp = 1.0;
	} else if (alpha <= 0.9) {
		tmp = t_0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 - (0.5 * alpha)) / 2.0
	tmp = 0
	if alpha <= -1.9e-114:
		tmp = t_0
	elif alpha <= -5.8e-170:
		tmp = 1.0
	elif alpha <= 0.9:
		tmp = t_0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 - Float64(0.5 * alpha)) / 2.0)
	tmp = 0.0
	if (alpha <= -1.9e-114)
		tmp = t_0;
	elseif (alpha <= -5.8e-170)
		tmp = 1.0;
	elseif (alpha <= 0.9)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	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 <= -1.9e-114)
		tmp = t_0;
	elseif (alpha <= -5.8e-170)
		tmp = 1.0;
	elseif (alpha <= 0.9)
		tmp = t_0;
	else
		tmp = (2.0 / alpha) / 2.0;
	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, -1.9e-114], t$95$0, If[LessEqual[alpha, -5.8e-170], 1.0, If[LessEqual[alpha, 0.9], t$95$0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -1.8999999999999999e-114 or -5.8000000000000001e-170 < alpha < 0.900000000000000022

   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 76.3%

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

      1. +-commutative 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in alpha around 0 76.1%

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

      1. *-commutative 76.1%

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

   9. Simplified 76.1%

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

   if -1.8999999999999999e-114 < alpha < -5.8000000000000001e-170

   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 76.1%

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

   if 0.900000000000000022 < alpha

   1. Initial program 25.7%

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

      1. +-commutative 25.7%

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

   3. Simplified 25.7%

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

   4. Taylor expanded in beta around 0 8.8%

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

      1. +-commutative 8.8%

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

   6. Simplified 8.8%

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

   7. Taylor expanded in alpha around inf 68.0%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -1.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{1 - 0.5 \cdot \alpha}{2}\\ \mathbf{elif}\;\alpha \leq -5.8 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.9:\\ \;\;\;\;\frac{1 - 0.5 \cdot \alpha}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - 0.5 \cdot \alpha}{2}\\ \mathbf{if}\;\alpha \leq -1.9 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq -2.65 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (* 0.5 alpha)) 2.0)))
   (if (<= alpha -1.9e-114)
     t_0
     (if (<= alpha -2.65e-169)
       1.0
       (if (<= alpha 2.0) t_0 (/ (/ (+ beta 2.0) alpha) 2.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 - (0.5 * alpha)) / 2.0;
	double tmp;
	if (alpha <= -1.9e-114) {
		tmp = t_0;
	} else if (alpha <= -2.65e-169) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = t_0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - (0.5d0 * alpha)) / 2.0d0
    if (alpha <= (-1.9d-114)) then
        tmp = t_0
    else if (alpha <= (-2.65d-169)) then
        tmp = 1.0d0
    else if (alpha <= 2.0d0) then
        tmp = t_0
    else
        tmp = ((beta + 2.0d0) / alpha) / 2.0d0
    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 <= -1.9e-114) {
		tmp = t_0;
	} else if (alpha <= -2.65e-169) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = t_0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 - (0.5 * alpha)) / 2.0
	tmp = 0
	if alpha <= -1.9e-114:
		tmp = t_0
	elif alpha <= -2.65e-169:
		tmp = 1.0
	elif alpha <= 2.0:
		tmp = t_0
	else:
		tmp = ((beta + 2.0) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 - Float64(0.5 * alpha)) / 2.0)
	tmp = 0.0
	if (alpha <= -1.9e-114)
		tmp = t_0;
	elseif (alpha <= -2.65e-169)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(beta + 2.0) / alpha) / 2.0);
	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 <= -1.9e-114)
		tmp = t_0;
	elseif (alpha <= -2.65e-169)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = t_0;
	else
		tmp = ((beta + 2.0) / alpha) / 2.0;
	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, -1.9e-114], t$95$0, If[LessEqual[alpha, -2.65e-169], 1.0, If[LessEqual[alpha, 2.0], t$95$0, N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -1.8999999999999999e-114 or -2.65e-169 < alpha < 2

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in beta around 0 76.3%

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

      1. +-commutative 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in alpha around 0 76.1%

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

      1. *-commutative 76.1%

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

   9. Simplified 76.1%

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

   if -1.8999999999999999e-114 < alpha < -2.65e-169

   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 76.1%

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

   if 2 < alpha

   1. Initial program 25.7%

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

      1. +-commutative 25.7%

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

   3. Simplified 25.7%

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

      1. div-sub 25.7%

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

      2. associate-+l- 27.4%

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

      3. associate-+l+ 27.4%

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

      4. associate-+l+ 27.4%

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

   5. Applied egg-rr 27.4%

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

   6. Taylor expanded in alpha around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. distribute-lft-in 81.5%

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

      3. neg-mul-1 81.5%

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

      4. metadata-eval 81.5%

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

   8. Simplified 81.5%

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

   9. Taylor expanded in alpha around 0 69.9%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -1.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{1 - 0.5 \cdot \alpha}{2}\\ \mathbf{elif}\;\alpha \leq -2.65 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;\frac{1 - 0.5 \cdot \alpha}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 67.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq -1.9 \cdot 10^{-114}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -2.6 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha -1.9e-114)
   0.5
   (if (<= alpha -2.6e-169)
     1.0
     (if (<= alpha 2.0) 0.5 (/ (/ 2.0 alpha) 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -1.9e-114) {
		tmp = 0.5;
	} else if (alpha <= -2.6e-169) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= (-1.9d-114)) then
        tmp = 0.5d0
    else if (alpha <= (-2.6d-169)) then
        tmp = 1.0d0
    else if (alpha <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -1.9e-114) {
		tmp = 0.5;
	} else if (alpha <= -2.6e-169) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= -1.9e-114:
		tmp = 0.5
	elif alpha <= -2.6e-169:
		tmp = 1.0
	elif alpha <= 2.0:
		tmp = 0.5
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= -1.9e-114)
		tmp = 0.5;
	elseif (alpha <= -2.6e-169)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= -1.9e-114)
		tmp = 0.5;
	elseif (alpha <= -2.6e-169)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = 0.5;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, -1.9e-114], 0.5, If[LessEqual[alpha, -2.6e-169], 1.0, If[LessEqual[alpha, 2.0], 0.5, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -1.8999999999999999e-114 or -2.60000000000000014e-169 < alpha < 2

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 73.8%

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

      2. clear-num 73.8%

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

      3. sub-neg 73.8%

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

      4. associate-+l+ 73.8%

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

      5. metadata-eval 73.8%

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

      6. metadata-eval 73.8%

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

      7. sub-neg 73.8%

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

      8. pow2 73.8%

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

      9. associate-+l+ 73.8%

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

      10. metadata-eval 73.8%

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

   5. Applied egg-rr 73.8%

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

   6. Taylor expanded in alpha around -inf 85.7%

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

   7. Taylor expanded in beta around 0 76.3%

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

   8. Taylor expanded in alpha around 0 75.4%

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

   if -1.8999999999999999e-114 < alpha < -2.60000000000000014e-169

   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 76.1%

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

   if 2 < alpha

   1. Initial program 25.7%

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

      1. +-commutative 25.7%

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

   3. Simplified 25.7%

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

   4. Taylor expanded in beta around 0 8.8%

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

      1. +-commutative 8.8%

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

   6. Simplified 8.8%

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

   7. Taylor expanded in alpha around inf 68.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -1.9 \cdot 10^{-114}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -2.6 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 10: 88.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 14.5:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 14.5)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta 2.0) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 14.5) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 14.5d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + 2.0d0) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 14.5) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 14.5:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + 2.0) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 14.5)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + 2.0) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 14.5)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + 2.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 14.5], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 14.5

   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 99.2%

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

   if 14.5 < alpha

   1. Initial program 25.7%

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

      1. +-commutative 25.7%

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

   3. Simplified 25.7%

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

      1. div-sub 25.7%

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

      2. associate-+l- 27.4%

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

      3. associate-+l+ 27.4%

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

      4. associate-+l+ 27.4%

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

   5. Applied egg-rr 27.4%

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

   6. Taylor expanded in alpha around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. distribute-lft-in 81.5%

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

      3. neg-mul-1 81.5%

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

      4. metadata-eval 81.5%

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

   8. Simplified 81.5%

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

   9. Taylor expanded in alpha around 0 69.9%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 14.5:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 11: 92.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.62 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{1 + 0.5 \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.62e-24)
   (/ (/ 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 <= 1.62e-24) {
		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 <= 1.62d-24) 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 <= 1.62e-24) {
		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 <= 1.62e-24:
		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 <= 1.62e-24)
		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 <= 1.62e-24)
		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, 1.62e-24], 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 < 1.62e-24

   1. Initial program 67.1%

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

      1. +-commutative 67.1%

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

   3. Simplified 67.1%

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

      1. flip-+ 67.1%

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

      2. clear-num 67.1%

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

      3. sub-neg 67.1%

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

      4. associate-+l+ 67.1%

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

      5. metadata-eval 67.1%

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

      6. metadata-eval 67.1%

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

      7. sub-neg 67.1%

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

      8. pow2 67.1%

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

      9. associate-+l+ 67.1%

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

      10. metadata-eval 67.1%

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

   5. Applied egg-rr 67.1%

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

   6. Taylor expanded in alpha around -inf 100.0%

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

   7. Taylor expanded in beta around 0 99.1%

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

   if 1.62e-24 < beta

   1. Initial program 86.4%

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

      1. +-commutative 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in alpha around 0 83.7%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.62 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{1 + 0.5 \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \]

Alternative 12: 70.8% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.05d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.0499999999999998

   1. Initial program 67.3%

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

      1. +-commutative 67.3%

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

   3. Simplified 67.3%

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

      1. flip-+ 67.3%

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

      2. clear-num 67.3%

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

      3. sub-neg 67.3%

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

      4. associate-+l+ 67.3%

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

      5. metadata-eval 67.3%

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

      6. metadata-eval 67.3%

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

      7. sub-neg 67.3%

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

      8. pow2 67.3%

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

      9. associate-+l+ 67.3%

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

      10. metadata-eval 67.3%

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

   5. Applied egg-rr 67.3%

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

   6. Taylor expanded in alpha around -inf 99.9%

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

   7. Taylor expanded in beta around 0 97.6%

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

   8. Taylor expanded in alpha around 0 63.3%

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

   if 2.0499999999999998 < beta

   1. Initial program 87.6%

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

      1. +-commutative 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in beta around inf 83.1%

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

4. Final simplification 69.4%

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

Alternative 13: 48.8% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 73.6%

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

   1. +-commutative 73.6%

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

3. Simplified 73.6%

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

   1. flip-+ 49.2%

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

   2. clear-num 49.2%

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

   3. sub-neg 49.2%

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

   4. associate-+l+ 49.2%

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

   5. metadata-eval 49.2%

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

   6. metadata-eval 49.2%

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

   7. sub-neg 49.2%

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

   8. pow2 49.2%

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

   9. associate-+l+ 49.2%

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

   10. metadata-eval 49.2%

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

5. Applied egg-rr 49.2%

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

6. Taylor expanded in alpha around -inf 82.7%

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

7. Taylor expanded in beta around 0 71.9%

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

8. Taylor expanded in alpha around 0 49.0%

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

9. Final simplification 49.0%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023192 
(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))