Octave 3.8, jcobi/1

Percentage Accurate: 75.8% → 99.3%

Time: 13.9s

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: 75.8% 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: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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))) < -1

   1. Initial program 5.5%

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

      1. +-commutative 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in alpha around inf 100.0%

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

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. div-inv 99.7%

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

      2. fma-define 99.7%

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

      3. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 99.8%

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

Alternative 2: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\beta}{\beta + 2}\\ \frac{\frac{1}{\frac{1}{t\_0} + \frac{\alpha \cdot \left(\frac{1}{\beta + 2} + \frac{\beta}{{\left(\beta + 2\right)}^{2}}\right)}{{t\_0}^{2}}}}{2} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ beta (+ beta 2.0)))))
   (/
    (/
     1.0
     (+
      (/ 1.0 t_0)
      (/
       (* alpha (+ (/ 1.0 (+ beta 2.0)) (/ beta (pow (+ beta 2.0) 2.0))))
       (pow t_0 2.0))))
    2.0)))

C

double code(double alpha, double beta) {
	double t_0 = 1.0 + (beta / (beta + 2.0));
	return (1.0 / ((1.0 / t_0) + ((alpha * ((1.0 / (beta + 2.0)) + (beta / pow((beta + 2.0), 2.0)))) / pow(t_0, 2.0)))) / 2.0;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = 1.0d0 + (beta / (beta + 2.0d0))
    code = (1.0d0 / ((1.0d0 / t_0) + ((alpha * ((1.0d0 / (beta + 2.0d0)) + (beta / ((beta + 2.0d0) ** 2.0d0)))) / (t_0 ** 2.0d0)))) / 2.0d0
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 1.0 + (beta / (beta + 2.0));
	return (1.0 / ((1.0 / t_0) + ((alpha * ((1.0 / (beta + 2.0)) + (beta / Math.pow((beta + 2.0), 2.0)))) / Math.pow(t_0, 2.0)))) / 2.0;
}

Python

def code(alpha, beta):
	t_0 = 1.0 + (beta / (beta + 2.0))
	return (1.0 / ((1.0 / t_0) + ((alpha * ((1.0 / (beta + 2.0)) + (beta / math.pow((beta + 2.0), 2.0)))) / math.pow(t_0, 2.0)))) / 2.0

Julia

function code(alpha, beta)
	t_0 = Float64(1.0 + Float64(beta / Float64(beta + 2.0)))
	return Float64(Float64(1.0 / Float64(Float64(1.0 / t_0) + Float64(Float64(alpha * Float64(Float64(1.0 / Float64(beta + 2.0)) + Float64(beta / (Float64(beta + 2.0) ^ 2.0)))) / (t_0 ^ 2.0)))) / 2.0)
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = 1.0 + (beta / (beta + 2.0));
	tmp = (1.0 / ((1.0 / t_0) + ((alpha * ((1.0 / (beta + 2.0)) + (beta / ((beta + 2.0) ^ 2.0)))) / (t_0 ^ 2.0)))) / 2.0;
end

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. +-commutative 76.9%

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

3. Simplified 76.9%

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

   1. add-log-exp 76.9%

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

   2. associate-+l+ 76.9%

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

6. Applied egg-rr 76.9%

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

   1. rem-log-exp 76.9%

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

   2. +-commutative 76.9%

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

   3. flip-+ 47.7%

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

   4. metadata-eval 47.7%

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

   5. unpow2 47.7%

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

   6. clear-num 47.7%

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

   7. clear-num 47.7%

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

   8. metadata-eval 47.7%

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

   9. unpow2 47.7%

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

   10. flip-+ 76.8%

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

   11. +-commutative 76.8%

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

8. Applied egg-rr 76.8%

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

9. Taylor expanded in alpha around 0 98.4%

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

10. Final simplification 98.4%

    \[\leadsto \frac{\frac{1}{\frac{1}{1 + \frac{\beta}{\beta + 2}} + \frac{\alpha \cdot \left(\frac{1}{\beta + 2} + \frac{\beta}{{\left(\beta + 2\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{\beta + 2}\right)}^{2}}}}{2} \]
11. Add Preprocessing

Alternative 3: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -1

   1. Initial program 5.5%

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

      1. +-commutative 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in alpha around inf 100.0%

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

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

4. Final simplification 99.7%

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

Alternative 4: 72.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq 1.3 \cdot 10^{-138}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)))
   (if (<= beta 1.3e-138)
     t_0
     (if (<= beta 1.2e-125)
       (/ (/ 2.0 alpha) 2.0)
       (if (<= beta 2.0) t_0 (/ (+ 2.0 (/ -2.0 beta)) 2.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= 1.3e-138) {
		tmp = t_0;
	} else if (beta <= 1.2e-125) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 + (-2.0 / beta)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    if (beta <= 1.3d-138) then
        tmp = t_0
    else if (beta <= 1.2d-125) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = (2.0d0 + ((-2.0d0) / beta)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= 1.3e-138) {
		tmp = t_0;
	} else if (beta <= 1.2e-125) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 + (-2.0 / beta)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (beta * 0.5)) / 2.0
	tmp = 0
	if beta <= 1.3e-138:
		tmp = t_0
	elif beta <= 1.2e-125:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = (2.0 + (-2.0 / beta)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0)
	tmp = 0.0
	if (beta <= 1.3e-138)
		tmp = t_0;
	elseif (beta <= 1.2e-125)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 + Float64(-2.0 / beta)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta * 0.5)) / 2.0;
	tmp = 0.0;
	if (beta <= 1.3e-138)
		tmp = t_0;
	elseif (beta <= 1.2e-125)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = (2.0 + (-2.0 / beta)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.3e-138], t$95$0, If[LessEqual[beta, 1.2e-125], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 2.0], t$95$0, N[(N[(2.0 + N[(-2.0 / beta), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.3e-138 or 1.2000000000000001e-125 < beta < 2

   1. Initial program 73.5%

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

      1. +-commutative 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in alpha around 0 69.8%

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

   6. Taylor expanded in beta around 0 68.8%

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

      1. *-commutative 68.8%

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

   8. Simplified 68.8%

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

   if 1.3e-138 < beta < 1.2000000000000001e-125

   1. Initial program 17.5%

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

      1. +-commutative 17.5%

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

   3. Simplified 17.5%

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

      1. add-log-exp 17.5%

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

      2. associate-+l+ 17.5%

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

   6. Applied egg-rr 17.5%

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

   7. Taylor expanded in beta around 0 17.5%

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

      1. +-commutative 17.5%

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

   9. Simplified 17.5%

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

   10. Taylor expanded in alpha around inf 87.7%

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

   if 2 < beta

   1. Initial program 88.1%

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

      1. +-commutative 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in beta around -inf 86.9%

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

      1. associate--l+ 86.9%

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

      2. associate--r+ 86.9%

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

      3. associate-*r/ 86.9%

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

      4. associate-*r/ 86.9%

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

      5. metadata-eval 86.9%

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

      6. div-sub 86.9%

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

      7. div-sub 86.9%

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

      8. sub-neg 86.9%

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

      9. mul-1-neg 86.9%

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

      10. distribute-neg-in 86.9%

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

      11. +-commutative 86.9%

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

      12. distribute-neg-in 86.9%

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

      13. metadata-eval 86.9%

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

      14. sub-neg 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in alpha around 0 86.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{-138}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{-140}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq 5.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* alpha -0.5)) 2.0)))
   (if (<= beta 4e-140)
     t_0
     (if (<= beta 5.8e-126)
       (/ (/ 2.0 alpha) 2.0)
       (if (<= beta 4.8e-26) t_0 (/ (+ 2.0 (/ -2.0 beta)) 2.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (alpha * -0.5)) / 2.0;
	double tmp;
	if (beta <= 4e-140) {
		tmp = t_0;
	} else if (beta <= 5.8e-126) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 4.8e-26) {
		tmp = t_0;
	} else {
		tmp = (2.0 + (-2.0 / beta)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (alpha * (-0.5d0))) / 2.0d0
    if (beta <= 4d-140) then
        tmp = t_0
    else if (beta <= 5.8d-126) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 4.8d-26) then
        tmp = t_0
    else
        tmp = (2.0d0 + ((-2.0d0) / beta)) / 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 (beta <= 4e-140) {
		tmp = t_0;
	} else if (beta <= 5.8e-126) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 4.8e-26) {
		tmp = t_0;
	} else {
		tmp = (2.0 + (-2.0 / beta)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (alpha * -0.5)) / 2.0
	tmp = 0
	if beta <= 4e-140:
		tmp = t_0
	elif beta <= 5.8e-126:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 4.8e-26:
		tmp = t_0
	else:
		tmp = (2.0 + (-2.0 / beta)) / 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 (beta <= 4e-140)
		tmp = t_0;
	elseif (beta <= 5.8e-126)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 4.8e-26)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 + Float64(-2.0 / beta)) / 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 (beta <= 4e-140)
		tmp = t_0;
	elseif (beta <= 5.8e-126)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 4.8e-26)
		tmp = t_0;
	else
		tmp = (2.0 + (-2.0 / beta)) / 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[beta, 4e-140], t$95$0, If[LessEqual[beta, 5.8e-126], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 4.8e-26], t$95$0, N[(N[(2.0 + N[(-2.0 / beta), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.9999999999999999e-140 or 5.79999999999999975e-126 < beta < 4.8000000000000002e-26

   1. Initial program 75.7%

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

      1. +-commutative 75.7%

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

   3. Simplified 75.7%

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

      1. add-log-exp 75.7%

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

      2. associate-+l+ 75.7%

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

   6. Applied egg-rr 75.7%

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

   7. Taylor expanded in beta around 0 74.3%

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

      1. +-commutative 74.3%

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

   9. Simplified 74.3%

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

   10. Taylor expanded in alpha around 0 70.4%

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

       1. *-commutative 70.4%

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

   12. Simplified 70.4%

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

   if 3.9999999999999999e-140 < beta < 5.79999999999999975e-126

   1. Initial program 17.5%

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

      1. +-commutative 17.5%

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

   3. Simplified 17.5%

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

      1. add-log-exp 17.5%

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

      2. associate-+l+ 17.5%

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

   6. Applied egg-rr 17.5%

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

   7. Taylor expanded in beta around 0 17.5%

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

      1. +-commutative 17.5%

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

   9. Simplified 17.5%

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

   10. Taylor expanded in alpha around inf 87.7%

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

   if 4.8000000000000002e-26 < beta

   1. Initial program 83.7%

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

      1. +-commutative 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in beta around -inf 82.3%

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

      1. associate--l+ 82.3%

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

      2. associate--r+ 82.3%

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

      3. associate-*r/ 82.3%

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

      4. associate-*r/ 82.3%

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

      5. metadata-eval 82.3%

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

      6. div-sub 82.3%

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

      7. div-sub 82.3%

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

      8. sub-neg 82.3%

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

      9. mul-1-neg 82.3%

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

      10. distribute-neg-in 82.3%

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

      11. +-commutative 82.3%

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

      12. distribute-neg-in 82.3%

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

      13. metadata-eval 82.3%

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

      14. sub-neg 82.3%

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

   7. Simplified 82.3%

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

   8. Taylor expanded in alpha around 0 82.4%

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

Alternative 6: 70.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{if}\;\beta \leq 4.2 \cdot 10^{-138}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq 1.1 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* alpha -0.5)) 2.0)))
   (if (<= beta 4.2e-138)
     t_0
     (if (<= beta 1.1e-124)
       (/ (/ 2.0 alpha) 2.0)
       (if (<= beta 4.8e-26) t_0 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (alpha * -0.5)) / 2.0;
	double tmp;
	if (beta <= 4.2e-138) {
		tmp = t_0;
	} else if (beta <= 1.1e-124) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 4.8e-26) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (alpha * (-0.5d0))) / 2.0d0
    if (beta <= 4.2d-138) then
        tmp = t_0
    else if (beta <= 1.1d-124) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 4.8d-26) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (alpha * -0.5)) / 2.0;
	double tmp;
	if (beta <= 4.2e-138) {
		tmp = t_0;
	} else if (beta <= 1.1e-124) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 4.8e-26) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (alpha * -0.5)) / 2.0
	tmp = 0
	if beta <= 4.2e-138:
		tmp = t_0
	elif beta <= 1.1e-124:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 4.8e-26:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(alpha * -0.5)) / 2.0)
	tmp = 0.0
	if (beta <= 4.2e-138)
		tmp = t_0;
	elseif (beta <= 1.1e-124)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 4.8e-26)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (alpha * -0.5)) / 2.0;
	tmp = 0.0;
	if (beta <= 4.2e-138)
		tmp = t_0;
	elseif (beta <= 1.1e-124)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 4.8e-26)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(alpha * -0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, 4.2e-138], t$95$0, If[LessEqual[beta, 1.1e-124], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 4.8e-26], t$95$0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.19999999999999972e-138 or 1.0999999999999999e-124 < beta < 4.8000000000000002e-26

   1. Initial program 75.7%

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

      1. +-commutative 75.7%

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

   3. Simplified 75.7%

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

      1. add-log-exp 75.7%

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

      2. associate-+l+ 75.7%

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

   6. Applied egg-rr 75.7%

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

   7. Taylor expanded in beta around 0 74.3%

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

      1. +-commutative 74.3%

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

   9. Simplified 74.3%

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

   10. Taylor expanded in alpha around 0 70.4%

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

       1. *-commutative 70.4%

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

   12. Simplified 70.4%

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

   if 4.19999999999999972e-138 < beta < 1.0999999999999999e-124

   1. Initial program 17.5%

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

      1. +-commutative 17.5%

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

   3. Simplified 17.5%

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

      1. add-log-exp 17.5%

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

      2. associate-+l+ 17.5%

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

   6. Applied egg-rr 17.5%

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

   7. Taylor expanded in beta around 0 17.5%

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

      1. +-commutative 17.5%

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

   9. Simplified 17.5%

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

   10. Taylor expanded in alpha around inf 87.7%

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

   if 4.8000000000000002e-26 < beta

   1. Initial program 83.7%

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

      1. +-commutative 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in beta around inf 82.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{-138}:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{elif}\;\beta \leq 1.1 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{1 + \alpha \cdot 0.5}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.3e-5)
   (/ (/ 1.0 (+ 1.0 (* alpha 0.5))) 2.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3e-5) {
		tmp = (1.0 / (1.0 + (alpha * 0.5))) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.3d-5) then
        tmp = (1.0d0 / (1.0d0 + (alpha * 0.5d0))) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3e-5) {
		tmp = (1.0 / (1.0 + (alpha * 0.5))) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.3e-5:
		tmp = (1.0 / (1.0 + (alpha * 0.5))) / 2.0
	else:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.3e-5)
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(alpha * 0.5))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.3e-5)
		tmp = (1.0 / (1.0 + (alpha * 0.5))) / 2.0;
	else
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.3e-5], N[(N[(1.0 / N[(1.0 + N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.29999999999999992e-5

   1. Initial program 70.8%

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

      1. +-commutative 70.8%

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

   3. Simplified 70.8%

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

      1. add-log-exp 70.8%

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

      2. associate-+l+ 70.8%

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

   6. Applied egg-rr 70.8%

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

      1. rem-log-exp 70.8%

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

      2. +-commutative 70.8%

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

      3. flip-+ 70.8%

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

      4. metadata-eval 70.8%

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

      5. unpow2 70.8%

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

      6. clear-num 70.8%

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

      7. clear-num 70.8%

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

      8. metadata-eval 70.8%

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

      9. unpow2 70.8%

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

      10. flip-+ 70.8%

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

      11. +-commutative 70.8%

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

   8. Applied egg-rr 70.8%

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

   9. Taylor expanded in alpha around 0 99.9%

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

   10. Taylor expanded in beta around 0 97.8%

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

   if 1.29999999999999992e-5 < beta

   1. Initial program 88.1%

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

      1. +-commutative 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in alpha around 0 87.4%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{1 + \alpha \cdot 0.5}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 14000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 14000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 14000.0) {
		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 <= 14000.0d0) 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 <= 14000.0) {
		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 <= 14000.0:
		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 <= 14000.0)
		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 <= 14000.0)
		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, 14000.0], 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 < 14000

   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. Taylor expanded in alpha around 0 97.5%

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

   if 14000 < alpha

   1. Initial program 21.2%

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

      1. +-commutative 21.2%

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

   3. Simplified 21.2%

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

      1. add-log-exp 21.3%

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

      2. associate-+l+ 21.3%

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

   6. Applied egg-rr 21.3%

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

   7. Taylor expanded in beta around 0 7.1%

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

      1. +-commutative 7.1%

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

   9. Simplified 7.1%

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

   10. Taylor expanded in alpha around inf 68.9%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 14000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 14000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 14000.0) 1.0 (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 14000.0) {
		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 <= 14000.0d0) 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 <= 14000.0) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 14000.0:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 14000.0)
		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 <= 14000.0)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 14000.0], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 14000

   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. Taylor expanded in beta around inf 48.4%

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

   if 14000 < alpha

   1. Initial program 21.2%

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

      1. +-commutative 21.2%

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

   3. Simplified 21.2%

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

      1. add-log-exp 21.3%

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

      2. associate-+l+ 21.3%

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

   6. Applied egg-rr 21.3%

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

   7. Taylor expanded in beta around 0 7.1%

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

      1. +-commutative 7.1%

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

   9. Simplified 7.1%

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

   10. Taylor expanded in alpha around inf 68.9%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 14000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.5% 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 76.9%

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

   1. +-commutative 76.9%

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

3. Simplified 76.9%

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

5. Taylor expanded in beta around inf 39.5%

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

6. Final simplification 39.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(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))