Octave 3.8, jcobi/1

Percentage Accurate: 75.1% → 99.9%

Time: 11.0s

Alternatives: 8

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.1% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \beta \cdot 2\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{t_0 \cdot \frac{1}{\alpha} + \frac{t_0 \cdot \frac{-1}{\alpha}}{\frac{\alpha}{\beta + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999998)
     (/
      (+
       (* t_0 (/ 1.0 alpha))
       (/ (* t_0 (/ -1.0 alpha)) (/ alpha (+ beta 2.0))))
      2.0)
     (/ (exp (log1p (/ (- beta alpha) (+ beta (+ alpha 2.0))))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta * 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998) {
		tmp = ((t_0 * (1.0 / alpha)) + ((t_0 * (-1.0 / alpha)) / (alpha / (beta + 2.0)))) / 2.0;
	} else {
		tmp = exp(log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta * 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998) {
		tmp = ((t_0 * (1.0 / alpha)) + ((t_0 * (-1.0 / alpha)) / (alpha / (beta + 2.0)))) / 2.0;
	} else {
		tmp = Math.exp(Math.log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 2.0 + (beta * 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998:
		tmp = ((t_0 * (1.0 / alpha)) + ((t_0 * (-1.0 / alpha)) / (alpha / (beta + 2.0)))) / 2.0
	else:
		tmp = math.exp(math.log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta * 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999998)
		tmp = Float64(Float64(Float64(t_0 * Float64(1.0 / alpha)) + Float64(Float64(t_0 * Float64(-1.0 / alpha)) / Float64(alpha / Float64(beta + 2.0)))) / 2.0);
	else
		tmp = Float64(exp(log1p(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999998], N[(N[(N[(t$95$0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * N[(-1.0 / alpha), $MachinePrecision]), $MachinePrecision] / N[(alpha / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[Log[1 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999998000000000054

   1. Initial program 8.4%

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

      1. associate-+l+ 8.7%

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

   3. Simplified 8.7%

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

      1. flip3-+ 8.6%

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

      2. add-exp-log 8.6%

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

      3. flip3-+ 8.7%

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

      4. +-commutative 8.7%

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

      5. log1p-udef 8.7%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}}}{2} \]

      6. +-commutative 8.7%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}\right)}}{2} \]

      7. associate-+l+ 8.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(2 + \alpha\right)}}\right)}}{2} \]

   5. Applied egg-rr 8.4%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(2 + \alpha\right)}\right)}}}{2} \]
   6. Step-by-step derivation

      1. +-commutative 8.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}\right)}}{2} \]

      2. +-commutative 8.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta}\right)}}{2} \]

   7. Simplified 8.4%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}}{2} \]

   8. Taylor expanded in alpha around inf 90.1%

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

      1. mul-1-neg 90.1%

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

      2. unsub-neg 90.1%

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

      3. exp-sum 90.4%

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

      4. rem-exp-log 90.4%

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

      5. sub-neg 90.4%

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

      6. mul-1-neg 90.4%

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

      7. remove-double-neg 90.4%

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

      8. +-commutative 90.4%

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

      9. associate-+r+ 90.4%

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

      10. count-2 90.4%

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

      11. +-commutative 90.4%

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

      12. rem-exp-log 99.5%

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

      13. associate-/l* 99.5%

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

   10. Simplified 99.5%

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

   if -0.999998000000000054 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

      1. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

      1. flip3-+ 99.8%

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

      2. add-exp-log 99.8%

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

      3. flip3-+ 99.8%

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

      4. +-commutative 99.8%

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

      5. log1p-udef 99.8%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}}}{2} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}\right)}}{2} \]

      7. associate-+l+ 99.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(2 + \alpha\right)}}\right)}}{2} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(2 + \alpha\right)}\right)}}}{2} \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}\right)}}{2} \]

      2. +-commutative 99.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta}\right)}}{2} \]

   7. Simplified 99.8%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\left(2 + \beta \cdot 2\right) \cdot \frac{1}{\alpha} + \frac{\left(2 + \beta \cdot 2\right) \cdot \frac{-1}{\alpha}}{\frac{\alpha}{\beta + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \beta \cdot 2\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{t_0 \cdot \frac{1}{\alpha} + \frac{t_0 \cdot \frac{-1}{\alpha}}{\frac{\alpha}{\beta + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999998)
     (/
      (+
       (* t_0 (/ 1.0 alpha))
       (/ (* t_0 (/ -1.0 alpha)) (/ alpha (+ beta 2.0))))
      2.0)
     (/ (+ 1.0 (/ (- beta alpha) (+ alpha (+ beta 2.0)))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta * 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998) {
		tmp = ((t_0 * (1.0 / alpha)) + ((t_0 * (-1.0 / alpha)) / (alpha / (beta + 2.0)))) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + (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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (beta * 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999998d0)) then
        tmp = ((t_0 * (1.0d0 / alpha)) + ((t_0 * ((-1.0d0) / alpha)) / (alpha / (beta + 2.0d0)))) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / (alpha + (beta + 2.0d0)))) / 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 tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998) {
		tmp = ((t_0 * (1.0 / alpha)) + ((t_0 * (-1.0 / alpha)) / (alpha / (beta + 2.0)))) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 2.0 + (beta * 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998:
		tmp = ((t_0 * (1.0 / alpha)) + ((t_0 * (-1.0 / alpha)) / (alpha / (beta + 2.0)))) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta * 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999998)
		tmp = Float64(Float64(Float64(t_0 * Float64(1.0 / alpha)) + Float64(Float64(t_0 * Float64(-1.0 / alpha)) / Float64(alpha / Float64(beta + 2.0)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta * 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998)
		tmp = ((t_0 * (1.0 / alpha)) + ((t_0 * (-1.0 / alpha)) / (alpha / (beta + 2.0)))) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999998], N[(N[(N[(t$95$0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * N[(-1.0 / alpha), $MachinePrecision]), $MachinePrecision] / N[(alpha / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999998000000000054

   1. Initial program 8.4%

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

      1. associate-+l+ 8.7%

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

   3. Simplified 8.7%

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

      1. flip3-+ 8.6%

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

      2. add-exp-log 8.6%

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

      3. flip3-+ 8.7%

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

      4. +-commutative 8.7%

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

      5. log1p-udef 8.7%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}}}{2} \]

      6. +-commutative 8.7%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}\right)}}{2} \]

      7. associate-+l+ 8.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(2 + \alpha\right)}}\right)}}{2} \]

   5. Applied egg-rr 8.4%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(2 + \alpha\right)}\right)}}}{2} \]
   6. Step-by-step derivation

      1. +-commutative 8.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}\right)}}{2} \]

      2. +-commutative 8.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta}\right)}}{2} \]

   7. Simplified 8.4%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}}{2} \]

   8. Taylor expanded in alpha around inf 90.1%

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

      1. mul-1-neg 90.1%

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

      2. unsub-neg 90.1%

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

      3. exp-sum 90.4%

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

      4. rem-exp-log 90.4%

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

      5. sub-neg 90.4%

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

      6. mul-1-neg 90.4%

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

      7. remove-double-neg 90.4%

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

      8. +-commutative 90.4%

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

      9. associate-+r+ 90.4%

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

      10. count-2 90.4%

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

      11. +-commutative 90.4%

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

      12. rem-exp-log 99.5%

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

      13. associate-/l* 99.5%

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

   10. Simplified 99.5%

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

   if -0.999998000000000054 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

      1. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\left(2 + \beta \cdot 2\right) \cdot \frac{1}{\alpha} + \frac{\left(2 + \beta \cdot 2\right) \cdot \frac{-1}{\alpha}}{\frac{\alpha}{\beta + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999996:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999996)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
   (/ (+ 1.0 (/ (- beta alpha) (+ alpha (+ beta 2.0)))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999996) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + (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 - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9999996d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / (alpha + (beta + 2.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999996) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999996:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9999996)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999996)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999996], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999959999999999

   1. Initial program 6.6%

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

      1. associate-+l+ 6.8%

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

   3. Simplified 6.8%

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

   4. Taylor expanded in alpha around inf 99.3%

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

   if -0.99999959999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.5%

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

      1. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999996:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \]

Alternative 4: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{if}\;\alpha \leq -4 \cdot 10^{-245}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq -4.2 \cdot 10^{-299}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.3:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* alpha -0.5)) 2.0)))
   (if (<= alpha -4e-245)
     t_0
     (if (<= alpha -4.2e-299)
       1.0
       (if (<= alpha 1.3) t_0 (/ (/ 2.0 alpha) 2.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (alpha * -0.5)) / 2.0;
	double tmp;
	if (alpha <= -4e-245) {
		tmp = t_0;
	} else if (alpha <= -4.2e-299) {
		tmp = 1.0;
	} else if (alpha <= 1.3) {
		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 + (alpha * (-0.5d0))) / 2.0d0
    if (alpha <= (-4d-245)) then
        tmp = t_0
    else if (alpha <= (-4.2d-299)) then
        tmp = 1.0d0
    else if (alpha <= 1.3d0) 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 + (alpha * -0.5)) / 2.0;
	double tmp;
	if (alpha <= -4e-245) {
		tmp = t_0;
	} else if (alpha <= -4.2e-299) {
		tmp = 1.0;
	} else if (alpha <= 1.3) {
		tmp = t_0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (alpha * -0.5)) / 2.0
	tmp = 0
	if alpha <= -4e-245:
		tmp = t_0
	elif alpha <= -4.2e-299:
		tmp = 1.0
	elif alpha <= 1.3:
		tmp = t_0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(alpha * -0.5)) / 2.0)
	tmp = 0.0
	if (alpha <= -4e-245)
		tmp = t_0;
	elseif (alpha <= -4.2e-299)
		tmp = 1.0;
	elseif (alpha <= 1.3)
		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 + (alpha * -0.5)) / 2.0;
	tmp = 0.0;
	if (alpha <= -4e-245)
		tmp = t_0;
	elseif (alpha <= -4.2e-299)
		tmp = 1.0;
	elseif (alpha <= 1.3)
		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[(alpha * -0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, -4e-245], t$95$0, If[LessEqual[alpha, -4.2e-299], 1.0, If[LessEqual[alpha, 1.3], t$95$0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -3.9999999999999997e-245 or -4.2000000000000002e-299 < alpha < 1.30000000000000004

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

      1. flip3-+ 99.9%

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

      2. add-exp-log 99.9%

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

      3. flip3-+ 100.0%

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

      4. +-commutative 100.0%

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

      5. log1p-udef 100.0%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}}}{2} \]

      6. +-commutative 100.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}\right)}}{2} \]

      7. associate-+l+ 100.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(2 + \alpha\right)}}\right)}}{2} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(2 + \alpha\right)}\right)}}}{2} \]
   6. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}\right)}}{2} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta}\right)}}{2} \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}}{2} \]

   8. Taylor expanded in beta around 0 82.1%

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

      1. log1p-def 82.1%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(-1 \cdot \frac{\alpha}{2 + \alpha}\right)}}}{2} \]

      2. associate-*r/ 82.1%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{-1 \cdot \alpha}{2 + \alpha}}\right)}}{2} \]

      3. mul-1-neg 82.1%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{-\alpha}}{2 + \alpha}\right)}}{2} \]

      4. +-commutative 82.1%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{-\alpha}{\color{blue}{\alpha + 2}}\right)}}{2} \]

   10. Simplified 82.1%

       \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{-\alpha}{\alpha + 2}\right)}}}{2} \]

   11. Taylor expanded in alpha around 0 81.3%

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

       1. *-commutative 81.3%

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

   13. Simplified 81.3%

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

   if -3.9999999999999997e-245 < alpha < -4.2000000000000002e-299

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in beta around inf 79.7%

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

   if 1.30000000000000004 < alpha

   1. Initial program 26.0%

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

      1. associate-+l+ 26.1%

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

   3. Simplified 26.1%

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

      1. flip3-+ 26.2%

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

      2. add-exp-log 26.2%

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

      3. flip3-+ 26.1%

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

      4. +-commutative 26.1%

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

      5. log1p-udef 26.2%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}}}{2} \]

      6. +-commutative 26.2%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}\right)}}{2} \]

      7. associate-+l+ 26.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(2 + \alpha\right)}}\right)}}{2} \]

   5. Applied egg-rr 26.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(2 + \alpha\right)}\right)}}}{2} \]
   6. Step-by-step derivation

      1. +-commutative 26.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}\right)}}{2} \]

      2. +-commutative 26.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta}\right)}}{2} \]

   7. Simplified 26.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}}{2} \]

   8. Taylor expanded in beta around 0 8.9%

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

      1. log1p-def 8.9%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(-1 \cdot \frac{\alpha}{2 + \alpha}\right)}}}{2} \]

      2. associate-*r/ 8.9%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{-1 \cdot \alpha}{2 + \alpha}}\right)}}{2} \]

      3. mul-1-neg 8.9%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{-\alpha}}{2 + \alpha}\right)}}{2} \]

      4. +-commutative 8.9%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{-\alpha}{\color{blue}{\alpha + 2}}\right)}}{2} \]

   10. Simplified 8.9%

       \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{-\alpha}{\alpha + 2}\right)}}}{2} \]

   11. Taylor expanded in alpha around inf 59.9%

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

       1. exp-sum 60.1%

          \[\leadsto \frac{\color{blue}{e^{\log 2} \cdot e^{\log \left(\frac{1}{\alpha}\right)}}}{2} \]

       2. rem-exp-log 60.1%

          \[\leadsto \frac{\color{blue}{2} \cdot e^{\log \left(\frac{1}{\alpha}\right)}}{2} \]

       3. rem-exp-log 65.4%

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

       4. associate-*r/ 65.4%

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

       5. metadata-eval 65.4%

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

   13. Simplified 65.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -4 \cdot 10^{-245}:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{elif}\;\alpha \leq -4.2 \cdot 10^{-299}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.3:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 88.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7500:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7500.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7500.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} 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) :: tmp
    if (alpha <= 7500.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    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 <= 7500.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 7500.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 7500.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	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 <= 7500.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 7500.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7500

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 98.7%

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

   if 7500 < alpha

   1. Initial program 25.2%

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

      1. associate-+l+ 25.3%

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

   3. Simplified 25.3%

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

      1. flip3-+ 25.4%

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

      2. add-exp-log 25.4%

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

      3. flip3-+ 25.3%

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

      4. +-commutative 25.3%

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

      5. log1p-udef 25.3%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}}}{2} \]

      6. +-commutative 25.3%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}\right)}}{2} \]

      7. associate-+l+ 25.2%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(2 + \alpha\right)}}\right)}}{2} \]

   5. Applied egg-rr 25.2%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(2 + \alpha\right)}\right)}}}{2} \]
   6. Step-by-step derivation

      1. +-commutative 25.2%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}\right)}}{2} \]

      2. +-commutative 25.2%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta}\right)}}{2} \]

   7. Simplified 25.2%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}}{2} \]

   8. Taylor expanded in beta around 0 8.8%

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

      1. log1p-def 8.8%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(-1 \cdot \frac{\alpha}{2 + \alpha}\right)}}}{2} \]

      2. associate-*r/ 8.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{-1 \cdot \alpha}{2 + \alpha}}\right)}}{2} \]

      3. mul-1-neg 8.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{-\alpha}}{2 + \alpha}\right)}}{2} \]

      4. +-commutative 8.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{-\alpha}{\color{blue}{\alpha + 2}}\right)}}{2} \]

   10. Simplified 8.8%

       \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{-\alpha}{\alpha + 2}\right)}}}{2} \]

   11. Taylor expanded in alpha around inf 60.4%

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

       1. exp-sum 60.6%

          \[\leadsto \frac{\color{blue}{e^{\log 2} \cdot e^{\log \left(\frac{1}{\alpha}\right)}}}{2} \]

       2. rem-exp-log 60.6%

          \[\leadsto \frac{\color{blue}{2} \cdot e^{\log \left(\frac{1}{\alpha}\right)}}{2} \]

       3. rem-exp-log 66.0%

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

       4. associate-*r/ 66.0%

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

       5. metadata-eval 66.0%

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

   13. Simplified 66.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7500:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 93.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5100:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 5100.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5100.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (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 <= 5100.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (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 <= 5100.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 5100.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 5100.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 5100.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 5100.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5100

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 98.7%

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

   if 5100 < alpha

   1. Initial program 25.2%

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

      1. associate-+l+ 25.3%

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

   3. Simplified 25.3%

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

   4. Taylor expanded in alpha around inf 81.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5100:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 51.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3600.0) 1.0 (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3600.0) {
		tmp = 1.0;
	} 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) :: tmp
    if (alpha <= 3600.0d0) then
        tmp = 1.0d0
    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 <= 3600.0) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 3600.0:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3600.0)
		tmp = 1.0;
	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 <= 3600.0)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 3600.0], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3600

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in beta around inf 39.5%

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

   if 3600 < alpha

   1. Initial program 25.2%

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

      1. associate-+l+ 25.3%

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

   3. Simplified 25.3%

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

      1. flip3-+ 25.4%

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

      2. add-exp-log 25.4%

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

      3. flip3-+ 25.3%

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

      4. +-commutative 25.3%

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

      5. log1p-udef 25.3%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}}}{2} \]

      6. +-commutative 25.3%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}\right)}}{2} \]

      7. associate-+l+ 25.2%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(2 + \alpha\right)}}\right)}}{2} \]

   5. Applied egg-rr 25.2%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(2 + \alpha\right)}\right)}}}{2} \]
   6. Step-by-step derivation

      1. +-commutative 25.2%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}\right)}}{2} \]

      2. +-commutative 25.2%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta}\right)}}{2} \]

   7. Simplified 25.2%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}}{2} \]

   8. Taylor expanded in beta around 0 8.8%

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

      1. log1p-def 8.8%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(-1 \cdot \frac{\alpha}{2 + \alpha}\right)}}}{2} \]

      2. associate-*r/ 8.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{-1 \cdot \alpha}{2 + \alpha}}\right)}}{2} \]

      3. mul-1-neg 8.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{-\alpha}}{2 + \alpha}\right)}}{2} \]

      4. +-commutative 8.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{-\alpha}{\color{blue}{\alpha + 2}}\right)}}{2} \]

   10. Simplified 8.8%

       \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{-\alpha}{\alpha + 2}\right)}}}{2} \]

   11. Taylor expanded in alpha around inf 60.4%

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

       1. exp-sum 60.6%

          \[\leadsto \frac{\color{blue}{e^{\log 2} \cdot e^{\log \left(\frac{1}{\alpha}\right)}}}{2} \]

       2. rem-exp-log 60.6%

          \[\leadsto \frac{\color{blue}{2} \cdot e^{\log \left(\frac{1}{\alpha}\right)}}{2} \]

       3. rem-exp-log 66.0%

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

       4. associate-*r/ 66.0%

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

       5. metadata-eval 66.0%

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

   13. Simplified 66.0%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 36.9% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

double code(double alpha, double beta) {
	return 1.0;
}

Fortran

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

Java

public static double code(double alpha, double beta) {
	return 1.0;
}

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 1.0;
end

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 73.4%

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

   1. associate-+l+ 73.4%

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

3. Simplified 73.4%

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

4. Taylor expanded in beta around inf 33.0%

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

5. Final simplification 33.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023319 
(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))