Octave 3.8, jcobi/1

Percentage Accurate: 74.4% → 99.8%

Time: 10.8s

Alternatives: 10

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.4% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \beta \cdot 2\\ t_1 := \beta + \left(\alpha + 2\right)\\ t_2 := \frac{\beta - \alpha}{t\_1}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{-0.5 \cdot \left(\log \left(e^{\frac{\beta + 2}{\alpha}}\right) \cdot t\_0 - t\_0\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{t\_2}^{3} + 1}{\left({t\_2}^{2} + \frac{\alpha - \beta}{t\_1}\right) + 1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* beta 2.0)))
        (t_1 (+ beta (+ alpha 2.0)))
        (t_2 (/ (- beta alpha) t_1)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9998)
     (/ (* -0.5 (- (* (log (exp (/ (+ beta 2.0) alpha))) t_0) t_0)) alpha)
     (/
      (/
       (+ (pow t_2 3.0) 1.0)
       (+ (+ (pow t_2 2.0) (/ (- alpha beta) t_1)) 1.0))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta * 2.0);
	double t_1 = beta + (alpha + 2.0);
	double t_2 = (beta - alpha) / t_1;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998) {
		tmp = (-0.5 * ((log(exp(((beta + 2.0) / alpha))) * t_0) - t_0)) / alpha;
	} else {
		tmp = ((pow(t_2, 3.0) + 1.0) / ((pow(t_2, 2.0) + ((alpha - beta) / t_1)) + 1.0)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 2.0d0 + (beta * 2.0d0)
    t_1 = beta + (alpha + 2.0d0)
    t_2 = (beta - alpha) / t_1
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9998d0)) then
        tmp = ((-0.5d0) * ((log(exp(((beta + 2.0d0) / alpha))) * t_0) - t_0)) / alpha
    else
        tmp = (((t_2 ** 3.0d0) + 1.0d0) / (((t_2 ** 2.0d0) + ((alpha - beta) / t_1)) + 1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta * 2.0);
	double t_1 = beta + (alpha + 2.0);
	double t_2 = (beta - alpha) / t_1;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998) {
		tmp = (-0.5 * ((Math.log(Math.exp(((beta + 2.0) / alpha))) * t_0) - t_0)) / alpha;
	} else {
		tmp = ((Math.pow(t_2, 3.0) + 1.0) / ((Math.pow(t_2, 2.0) + ((alpha - beta) / t_1)) + 1.0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 2.0 + (beta * 2.0)
	t_1 = beta + (alpha + 2.0)
	t_2 = (beta - alpha) / t_1
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998:
		tmp = (-0.5 * ((math.log(math.exp(((beta + 2.0) / alpha))) * t_0) - t_0)) / alpha
	else:
		tmp = ((math.pow(t_2, 3.0) + 1.0) / ((math.pow(t_2, 2.0) + ((alpha - beta) / t_1)) + 1.0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta * 2.0))
	t_1 = Float64(beta + Float64(alpha + 2.0))
	t_2 = Float64(Float64(beta - alpha) / t_1)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9998)
		tmp = Float64(Float64(-0.5 * Float64(Float64(log(exp(Float64(Float64(beta + 2.0) / alpha))) * t_0) - t_0)) / alpha);
	else
		tmp = Float64(Float64(Float64((t_2 ^ 3.0) + 1.0) / Float64(Float64((t_2 ^ 2.0) + Float64(Float64(alpha - beta) / t_1)) + 1.0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta * 2.0);
	t_1 = beta + (alpha + 2.0);
	t_2 = (beta - alpha) / t_1;
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998)
		tmp = (-0.5 * ((log(exp(((beta + 2.0) / alpha))) * t_0) - t_0)) / alpha;
	else
		tmp = (((t_2 ^ 3.0) + 1.0) / (((t_2 ^ 2.0) + ((alpha - beta) / t_1)) + 1.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta - alpha), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9998], N[(N[(-0.5 * N[(N[(N[Log[N[Exp[N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(N[Power[t$95$2, 3.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[Power[t$95$2, 2.0], $MachinePrecision] + N[(N[(alpha - beta), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99980000000000002

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in alpha around -inf 92.8%

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

      1. associate-*r/ 92.8%

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

   7. Simplified 92.8%

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

   8. Taylor expanded in alpha around inf 92.8%

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

      1. +-commutative 92.8%

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

      2. +-commutative 92.8%

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

      3. count-2 92.8%

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

      4. associate-+r+ 92.8%

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

      5. *-commutative 92.8%

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

      6. associate-*r/ 99.7%

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

      7. *-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. count-2 99.7%

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

      11. +-commutative 99.7%

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

   10. Simplified 99.7%

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

       1. add-log-exp 99.7%

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

   12. Applied egg-rr 99.7%

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

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. flip3-+ 99.9%

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

      3. metadata-eval 99.9%

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

      4. div-inv 99.8%

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

      5. div-inv 99.9%

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

      6. associate-+l+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. pow2 99.9%

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

      9. associate-+l+ 99.9%

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

      10. *-un-lft-identity 99.9%

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

      11. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{-0.5 \cdot \left(\log \left(e^{\frac{\beta + 2}{\alpha}}\right) \cdot \left(2 + \beta \cdot 2\right) - \left(2 + \beta \cdot 2\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3} + 1}{\left({\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2} + \frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}\right) + 1}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ t_1 := 2 + \beta \cdot 2\\ \mathbf{if}\;t\_0 \leq -0.9998:\\ \;\;\;\;\frac{-0.5 \cdot \left(\log \left(e^{\frac{\beta + 2}{\alpha}}\right) \cdot t\_1 - t\_1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0)))
        (t_1 (+ 2.0 (* beta 2.0))))
   (if (<= t_0 -0.9998)
     (/ (* -0.5 (- (* (log (exp (/ (+ beta 2.0) alpha))) t_1) t_1)) alpha)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double t_1 = 2.0 + (beta * 2.0);
	double tmp;
	if (t_0 <= -0.9998) {
		tmp = (-0.5 * ((log(exp(((beta + 2.0) / alpha))) * t_1) - t_1)) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    t_1 = 2.0d0 + (beta * 2.0d0)
    if (t_0 <= (-0.9998d0)) then
        tmp = ((-0.5d0) * ((log(exp(((beta + 2.0d0) / alpha))) * t_1) - t_1)) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double t_1 = 2.0 + (beta * 2.0);
	double tmp;
	if (t_0 <= -0.9998) {
		tmp = (-0.5 * ((Math.log(Math.exp(((beta + 2.0) / alpha))) * t_1) - t_1)) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	t_1 = 2.0 + (beta * 2.0)
	tmp = 0
	if t_0 <= -0.9998:
		tmp = (-0.5 * ((math.log(math.exp(((beta + 2.0) / alpha))) * t_1) - t_1)) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	t_1 = Float64(2.0 + Float64(beta * 2.0))
	tmp = 0.0
	if (t_0 <= -0.9998)
		tmp = Float64(Float64(-0.5 * Float64(Float64(log(exp(Float64(Float64(beta + 2.0) / alpha))) * t_1) - t_1)) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	t_1 = 2.0 + (beta * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.9998)
		tmp = (-0.5 * ((log(exp(((beta + 2.0) / alpha))) * t_1) - t_1)) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9998], N[(N[(-0.5 * N[(N[(N[Log[N[Exp[N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99980000000000002

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in alpha around -inf 92.8%

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

      1. associate-*r/ 92.8%

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

   7. Simplified 92.8%

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

   8. Taylor expanded in alpha around inf 92.8%

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

      1. +-commutative 92.8%

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

      2. +-commutative 92.8%

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

      3. count-2 92.8%

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

      4. associate-+r+ 92.8%

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

      5. *-commutative 92.8%

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

      6. associate-*r/ 99.7%

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

      7. *-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. count-2 99.7%

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

      11. +-commutative 99.7%

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

   10. Simplified 99.7%

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

       1. add-log-exp 99.7%

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

   12. Applied egg-rr 99.7%

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

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{-0.5 \cdot \left(\log \left(e^{\frac{\beta + 2}{\alpha}}\right) \cdot \left(2 + \beta \cdot 2\right) - \left(2 + \beta \cdot 2\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha} \cdot \left(-1 - \beta\right) + \left(\beta + 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9998)
     (/ (+ (* (/ (+ beta 2.0) alpha) (- -1.0 beta)) (+ beta 1.0)) alpha)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9998) {
		tmp = ((((beta + 2.0) / alpha) * (-1.0 - beta)) + (beta + 1.0)) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.9998d0)) then
        tmp = ((((beta + 2.0d0) / alpha) * ((-1.0d0) - beta)) + (beta + 1.0d0)) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9998) {
		tmp = ((((beta + 2.0) / alpha) * (-1.0 - beta)) + (beta + 1.0)) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.9998:
		tmp = ((((beta + 2.0) / alpha) * (-1.0 - beta)) + (beta + 1.0)) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.9998)
		tmp = Float64(Float64(Float64(Float64(Float64(beta + 2.0) / alpha) * Float64(-1.0 - beta)) + Float64(beta + 1.0)) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.9998)
		tmp = ((((beta + 2.0) / alpha) * (-1.0 - beta)) + (beta + 1.0)) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9998], N[(N[(N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * N[(-1.0 - beta), $MachinePrecision]), $MachinePrecision] + N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99980000000000002

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in alpha around -inf 92.8%

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

      1. associate-*r/ 92.8%

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

   7. Simplified 92.8%

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

   8. Taylor expanded in alpha around inf 92.8%

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

      1. +-commutative 92.8%

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

      2. +-commutative 92.8%

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

      3. count-2 92.8%

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

      4. associate-+r+ 92.8%

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

      5. *-commutative 92.8%

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

      6. associate-*r/ 99.7%

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

      7. *-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. associate-+r+ 99.7%

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

      10. count-2 99.7%

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

      11. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in alpha around inf 92.8%

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

   13. Simplified 99.7%

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

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha} \cdot \left(-1 - \beta\right) + \left(\beta + 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999995) (/ (+ beta 1.0) alpha) (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99999995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.99999995d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99999995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.99999995:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.99999995)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.99999995)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99999995], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999949999999971

   1. Initial program 6.4%

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

      1. +-commutative 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in alpha around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. associate-*r* 99.6%

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

      5. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in alpha around 0 99.6%

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

   if -0.999999949999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \alpha \cdot -0.25\\ \mathbf{if}\;\alpha \leq -4 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq -2.2 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* alpha -0.25))))
   (if (<= alpha -4e-132)
     t_0
     (if (<= alpha -2.2e-195)
       1.0
       (if (<= alpha 2.0) t_0 (/ (+ beta 1.0) alpha))))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -4e-132) {
		tmp = t_0;
	} else if (alpha <= -2.2e-195) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (alpha * (-0.25d0))
    if (alpha <= (-4d-132)) then
        tmp = t_0
    else if (alpha <= (-2.2d-195)) then
        tmp = 1.0d0
    else if (alpha <= 2.0d0) then
        tmp = t_0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -4e-132) {
		tmp = t_0;
	} else if (alpha <= -2.2e-195) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (alpha * -0.25)
	tmp = 0
	if alpha <= -4e-132:
		tmp = t_0
	elif alpha <= -2.2e-195:
		tmp = 1.0
	elif alpha <= 2.0:
		tmp = t_0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(alpha * -0.25))
	tmp = 0.0
	if (alpha <= -4e-132)
		tmp = t_0;
	elseif (alpha <= -2.2e-195)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (alpha * -0.25);
	tmp = 0.0;
	if (alpha <= -4e-132)
		tmp = t_0;
	elseif (alpha <= -2.2e-195)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = t_0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, -4e-132], t$95$0, If[LessEqual[alpha, -2.2e-195], 1.0, If[LessEqual[alpha, 2.0], t$95$0, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -3.9999999999999999e-132 or -2.20000000000000005e-195 < alpha < 2

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 76.2%

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

      1. +-commutative 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in alpha around 0 75.7%

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

      1. *-commutative 75.7%

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

   10. Simplified 75.7%

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

   if -3.9999999999999999e-132 < alpha < -2.20000000000000005e-195

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 74.6%

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

   if 2 < alpha

   1. Initial program 21.8%

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

      1. +-commutative 21.8%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in alpha around inf 84.8%

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

      1. associate-*r/ 84.8%

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

      2. distribute-lft-in 84.8%

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

      3. metadata-eval 84.8%

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

      4. associate-*r* 84.8%

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

      5. metadata-eval 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in alpha around 0 84.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -4 \cdot 10^{-132}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq -2.2 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \alpha \cdot -0.25\\ \mathbf{if}\;\alpha \leq -3.3 \cdot 10^{-133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq -2 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.95:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* alpha -0.25))))
   (if (<= alpha -3.3e-133)
     t_0
     (if (<= alpha -2e-195) 1.0 (if (<= alpha 0.95) t_0 (/ 1.0 alpha))))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -3.3e-133) {
		tmp = t_0;
	} else if (alpha <= -2e-195) {
		tmp = 1.0;
	} else if (alpha <= 0.95) {
		tmp = t_0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (alpha * (-0.25d0))
    if (alpha <= (-3.3d-133)) then
        tmp = t_0
    else if (alpha <= (-2d-195)) then
        tmp = 1.0d0
    else if (alpha <= 0.95d0) then
        tmp = t_0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -3.3e-133) {
		tmp = t_0;
	} else if (alpha <= -2e-195) {
		tmp = 1.0;
	} else if (alpha <= 0.95) {
		tmp = t_0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (alpha * -0.25)
	tmp = 0
	if alpha <= -3.3e-133:
		tmp = t_0
	elif alpha <= -2e-195:
		tmp = 1.0
	elif alpha <= 0.95:
		tmp = t_0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(alpha * -0.25))
	tmp = 0.0
	if (alpha <= -3.3e-133)
		tmp = t_0;
	elseif (alpha <= -2e-195)
		tmp = 1.0;
	elseif (alpha <= 0.95)
		tmp = t_0;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (alpha * -0.25);
	tmp = 0.0;
	if (alpha <= -3.3e-133)
		tmp = t_0;
	elseif (alpha <= -2e-195)
		tmp = 1.0;
	elseif (alpha <= 0.95)
		tmp = t_0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, -3.3e-133], t$95$0, If[LessEqual[alpha, -2e-195], 1.0, If[LessEqual[alpha, 0.95], t$95$0, N[(1.0 / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -3.30000000000000009e-133 or -2.0000000000000002e-195 < alpha < 0.94999999999999996

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 76.2%

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

      1. +-commutative 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in alpha around 0 75.7%

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

      1. *-commutative 75.7%

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

   10. Simplified 75.7%

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

   if -3.30000000000000009e-133 < alpha < -2.0000000000000002e-195

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 74.6%

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

   if 0.94999999999999996 < alpha

   1. Initial program 21.8%

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

      1. +-commutative 21.8%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in alpha around inf 84.8%

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

      1. associate-*r/ 84.8%

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

      2. distribute-lft-in 84.8%

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

      3. metadata-eval 84.8%

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

      4. associate-*r* 84.8%

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

      5. metadata-eval 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in beta around 0 67.8%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -3.3 \cdot 10^{-133}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq -2 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.95:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq -2.2 \cdot 10^{-134}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -2.3 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.9:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha -2.2e-134)
   0.5
   (if (<= alpha -2.3e-195) 1.0 (if (<= alpha 0.9) 0.5 (/ 1.0 alpha)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -2.2e-134) {
		tmp = 0.5;
	} else if (alpha <= -2.3e-195) {
		tmp = 1.0;
	} else if (alpha <= 0.9) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= (-2.2d-134)) then
        tmp = 0.5d0
    else if (alpha <= (-2.3d-195)) then
        tmp = 1.0d0
    else if (alpha <= 0.9d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -2.2e-134) {
		tmp = 0.5;
	} else if (alpha <= -2.3e-195) {
		tmp = 1.0;
	} else if (alpha <= 0.9) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= -2.2e-134:
		tmp = 0.5
	elif alpha <= -2.3e-195:
		tmp = 1.0
	elif alpha <= 0.9:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= -2.2e-134)
		tmp = 0.5;
	elseif (alpha <= -2.3e-195)
		tmp = 1.0;
	elseif (alpha <= 0.9)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= -2.2e-134)
		tmp = 0.5;
	elseif (alpha <= -2.3e-195)
		tmp = 1.0;
	elseif (alpha <= 0.9)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, -2.2e-134], 0.5, If[LessEqual[alpha, -2.3e-195], 1.0, If[LessEqual[alpha, 0.9], 0.5, N[(1.0 / alpha), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -2.2e-134 or -2.3000000000000002e-195 < 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. Add Preprocessing

   5. Taylor expanded in beta around 0 76.2%

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

      1. +-commutative 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in alpha around 0 74.9%

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

   if -2.2e-134 < alpha < -2.3000000000000002e-195

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 74.6%

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

   if 0.900000000000000022 < alpha

   1. Initial program 21.8%

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

      1. +-commutative 21.8%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in alpha around inf 84.8%

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

      1. associate-*r/ 84.8%

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

      2. distribute-lft-in 84.8%

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

      3. metadata-eval 84.8%

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

      4. associate-*r* 84.8%

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

      5. metadata-eval 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in beta around 0 67.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -2.2 \cdot 10^{-134}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -2.3 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.9:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1020:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1020.0)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1020.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1020.0d0) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1020.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1020.0:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1020.0)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1020.0)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1020.0], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1020

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.8%

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

   if 1020 < alpha

   1. Initial program 21.8%

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

      1. +-commutative 21.8%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in alpha around inf 84.8%

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

      1. associate-*r/ 84.8%

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

      2. distribute-lft-in 84.8%

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

      3. metadata-eval 84.8%

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

      4. associate-*r* 84.8%

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

      5. metadata-eval 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in alpha around 0 84.8%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1020:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.56d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1.5600000000000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 72.5%

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

      1. +-commutative 72.5%

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

   7. Simplified 72.5%

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

   8. Taylor expanded in alpha around 0 71.3%

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

   if 1.5600000000000001 < alpha

   1. Initial program 21.8%

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

      1. +-commutative 21.8%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in alpha around inf 84.8%

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

      1. associate-*r/ 84.8%

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

      2. distribute-lft-in 84.8%

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

      3. metadata-eval 84.8%

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

      4. associate-*r* 84.8%

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

      5. metadata-eval 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in beta around 0 67.8%

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

Alternative 10: 49.4% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 70.1%

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

   1. +-commutative 70.1%

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

3. Simplified 70.1%

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

5. Taylor expanded in beta around 0 47.2%

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

   1. +-commutative 47.2%

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

7. Simplified 47.2%

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

8. Taylor expanded in alpha around 0 46.9%

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

Reproduce

herbie shell --seed 2024089 
(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))