Octave 3.8, jcobi/1

Percentage Accurate: 74.4% → 99.8%

Time: 11.5s

Alternatives: 10

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.998:\\ \;\;\;\;\frac{\frac{\left(\beta + \frac{\frac{8}{\alpha} + -4}{\alpha}\right) + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.998)
     (/
      (/ (+ (+ beta (/ (+ (/ 8.0 alpha) -4.0) alpha)) (- beta -2.0)) alpha)
      2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.998) {
		tmp = (((beta + (((8.0 / alpha) + -4.0) / alpha)) + (beta - -2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.998d0)) then
        tmp = (((beta + (((8.0d0 / alpha) + (-4.0d0)) / alpha)) + (beta - (-2.0d0))) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.998) {
		tmp = (((beta + (((8.0 / alpha) + -4.0) / alpha)) + (beta - -2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.998:
		tmp = (((beta + (((8.0 / alpha) + -4.0) / alpha)) + (beta - -2.0)) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.998)
		tmp = Float64(Float64(Float64(Float64(beta + Float64(Float64(Float64(8.0 / alpha) + -4.0) / alpha)) + Float64(beta - -2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.998)
		tmp = (((beta + (((8.0 / alpha) + -4.0) / alpha)) + (beta - -2.0)) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.998], N[(N[(N[(N[(beta + N[(N[(N[(8.0 / alpha), $MachinePrecision] + -4.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.998

   1. Initial program 8.1%

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

      1. +-commutative 8.1%

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

   3. Simplified 8.1%

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

   5. Taylor expanded in alpha around -inf 89.6%

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

   7. Simplified 89.6%

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

   8. Taylor expanded in beta around 0 99.5%

      \[\leadsto \frac{\frac{\left(-2 - \beta\right) - \left(\beta + \color{blue}{\frac{8 \cdot \frac{1}{\alpha} - 4}{\alpha}}\right)}{-\alpha}}{2} \]
   9. Step-by-step derivation

      1. sub-neg 99.5%

         \[\leadsto \frac{\frac{\left(-2 - \beta\right) - \left(\beta + \frac{\color{blue}{8 \cdot \frac{1}{\alpha} + \left(-4\right)}}{\alpha}\right)}{-\alpha}}{2} \]

      2. associate-*r/ 99.5%

         \[\leadsto \frac{\frac{\left(-2 - \beta\right) - \left(\beta + \frac{\color{blue}{\frac{8 \cdot 1}{\alpha}} + \left(-4\right)}{\alpha}\right)}{-\alpha}}{2} \]

      3. metadata-eval 99.5%

         \[\leadsto \frac{\frac{\left(-2 - \beta\right) - \left(\beta + \frac{\frac{\color{blue}{8}}{\alpha} + \left(-4\right)}{\alpha}\right)}{-\alpha}}{2} \]

      4. metadata-eval 99.5%

         \[\leadsto \frac{\frac{\left(-2 - \beta\right) - \left(\beta + \frac{\frac{8}{\alpha} + \color{blue}{-4}}{\alpha}\right)}{-\alpha}}{2} \]

   10. Simplified 99.5%

       \[\leadsto \frac{\frac{\left(-2 - \beta\right) - \left(\beta + \color{blue}{\frac{\frac{8}{\alpha} + -4}{\alpha}}\right)}{-\alpha}}{2} \]

   if -0.998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.998:\\ \;\;\;\;\frac{\frac{\left(\beta + \frac{\frac{8}{\alpha} + -4}{\alpha}\right) + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\beta - \alpha\right) \cdot \frac{-1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999998)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
   (/ (- 1.0 (* (- beta alpha) (/ -1.0 (+ beta (+ alpha 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999998) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 - ((beta - alpha) * (-1.0 / (beta + (alpha + 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)) <= (-0.9999998d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 - ((beta - alpha) * ((-1.0d0) / (beta + (alpha + 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)) <= -0.9999998) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 - ((beta - alpha) * (-1.0 / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999998:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (1.0 - ((beta - alpha) * (-1.0 / (beta + (alpha + 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)) <= -0.9999998)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(beta - alpha) * Float64(-1.0 / Float64(beta + Float64(alpha + 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)) <= -0.9999998)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = (1.0 - ((beta - alpha) * (-1.0 / (beta + (alpha + 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], -0.9999998], 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[(-1.0 / 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) #s(literal 2 binary64))) < -0.999999799999999994

   1. Initial program 6.1%

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

      1. +-commutative 6.1%

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

   3. Simplified 6.1%

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

   5. Taylor expanded in alpha around inf 99.8%

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

   if -0.999999799999999994 < (/.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. clear-num 99.6%

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

      2. associate-/r/ 99.7%

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

      3. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\beta - \alpha\right) \cdot \frac{-1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9999998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9999998)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9999998) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.9999998d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9999998) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.9999998:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.9999998)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.9999998)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9999998], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999799999999994

   1. Initial program 6.1%

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

      1. +-commutative 6.1%

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

   3. Simplified 6.1%

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

   5. Taylor expanded in alpha around inf 99.8%

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

   if -0.999999799999999994 < (/.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 200000000:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 5.2 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.75e-7)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (if (<= alpha 200000000.0)
     (/ (- 1.0 (/ alpha (+ alpha 2.0))) 2.0)
     (if (<= alpha 5.2e+22) 1.0 (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.75e-7) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 200000000.0) {
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	} else if (alpha <= 5.2e+22) {
		tmp = 1.0;
	} 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 <= 1.75d-7) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else if (alpha <= 200000000.0d0) then
        tmp = (1.0d0 - (alpha / (alpha + 2.0d0))) / 2.0d0
    else if (alpha <= 5.2d+22) then
        tmp = 1.0d0
    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 <= 1.75e-7) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 200000000.0) {
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	} else if (alpha <= 5.2e+22) {
		tmp = 1.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.75e-7:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	elif alpha <= 200000000.0:
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0
	elif alpha <= 5.2e+22:
		tmp = 1.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.75e-7)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	elseif (alpha <= 200000000.0)
		tmp = Float64(Float64(1.0 - Float64(alpha / Float64(alpha + 2.0))) / 2.0);
	elseif (alpha <= 5.2e+22)
		tmp = 1.0;
	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 <= 1.75e-7)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	elseif (alpha <= 200000000.0)
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	elseif (alpha <= 5.2e+22)
		tmp = 1.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.75e-7], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 200000000.0], N[(N[(1.0 - N[(alpha / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 5.2e+22], 1.0, N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 1.74999999999999992e-7

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 99.9%

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

   if 1.74999999999999992e-7 < alpha < 2e8

   1. Initial program 92.9%

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

      1. +-commutative 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in beta around 0 81.8%

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

      1. +-commutative 81.8%

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

   7. Simplified 81.8%

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

   if 2e8 < alpha < 5.2e22

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 100.0%

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

   if 5.2e22 < alpha

   1. Initial program 17.4%

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

      1. +-commutative 17.4%

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

   3. Simplified 17.4%

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

   5. Taylor expanded in alpha around inf 88.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 200000000:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 5.2 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.45 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 190000000:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 10^{+23}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.45e-7)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (if (<= alpha 190000000.0)
     (/ (- 1.0 (/ alpha (+ alpha 2.0))) 2.0)
     (if (<= alpha 1e+23) 1.0 (/ (/ 2.0 alpha) 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.45e-7) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 190000000.0) {
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	} else if (alpha <= 1e+23) {
		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 <= 1.45d-7) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else if (alpha <= 190000000.0d0) then
        tmp = (1.0d0 - (alpha / (alpha + 2.0d0))) / 2.0d0
    else if (alpha <= 1d+23) 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 <= 1.45e-7) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 190000000.0) {
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	} else if (alpha <= 1e+23) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.45e-7:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	elif alpha <= 190000000.0:
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0
	elif alpha <= 1e+23:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.45e-7)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	elseif (alpha <= 190000000.0)
		tmp = Float64(Float64(1.0 - Float64(alpha / Float64(alpha + 2.0))) / 2.0);
	elseif (alpha <= 1e+23)
		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 <= 1.45e-7)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	elseif (alpha <= 190000000.0)
		tmp = (1.0 - (alpha / (alpha + 2.0))) / 2.0;
	elseif (alpha <= 1e+23)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.45e-7], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 190000000.0], N[(N[(1.0 - N[(alpha / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1e+23], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 1.4499999999999999e-7

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 99.9%

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

   if 1.4499999999999999e-7 < alpha < 1.9e8

   1. Initial program 92.9%

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

      1. +-commutative 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in beta around 0 81.8%

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

      1. +-commutative 81.8%

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

   7. Simplified 81.8%

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

   if 1.9e8 < alpha < 9.9999999999999992e22

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 100.0%

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

   if 9.9999999999999992e22 < alpha

   1. Initial program 17.4%

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

      1. +-commutative 17.4%

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

   3. Simplified 17.4%

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

   5. Taylor expanded in beta around 0 5.3%

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

      1. +-commutative 5.3%

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

   7. Simplified 5.3%

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

   8. Taylor expanded in alpha around inf 67.2%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.45 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 190000000:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 10^{+23}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.3e+23)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.3e+23) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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 <= 2.3d+23) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 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 <= 2.3e+23) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.3e+23:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.3e+23)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 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 <= 2.3e+23)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.3e+23], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.3e23

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in alpha around 0 96.8%

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

   if 2.3e23 < alpha

   1. Initial program 17.4%

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

      1. +-commutative 17.4%

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

   3. Simplified 17.4%

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

   5. Taylor expanded in beta around 0 5.3%

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

      1. +-commutative 5.3%

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

   7. Simplified 5.3%

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

   8. Taylor expanded in alpha around inf 67.2%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = ((beta * 0.5) + 1.0) / 2.0;
	} 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) :: tmp
    if (beta <= 2.0d0) then
        tmp = ((beta * 0.5d0) + 1.0d0) / 2.0d0
    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 tmp;
	if (beta <= 2.0) {
		tmp = ((beta * 0.5) + 1.0) / 2.0;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 70.7%

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

      1. +-commutative 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in alpha around 0 67.2%

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

   6. Taylor expanded in beta around 0 66.7%

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

   if 2 < beta

   1. Initial program 84.4%

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

      1. +-commutative 84.4%

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

   3. Simplified 84.4%

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

      1. clear-num 84.4%

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

      2. associate-/r/ 84.4%

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

      3. associate-+l+ 84.4%

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

   6. Applied egg-rr 84.4%

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

   7. Taylor expanded in beta around inf 82.9%

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

      1. mul-1-neg 82.9%

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

      2. unsub-neg 82.9%

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

      3. *-commutative 82.9%

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

   9. Simplified 82.9%

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

   10. Taylor expanded in alpha around 0 83.1%

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

4. Final simplification 72.5%

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

Alternative 8: 71.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 70.7%

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

      1. +-commutative 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in alpha around 0 67.2%

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

   6. Taylor expanded in beta around 0 66.7%

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

   if 2 < beta

   1. Initial program 84.4%

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

      1. +-commutative 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in beta around inf 81.9%

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

4. Final simplification 72.0%

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

Alternative 9: 51.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.4 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 5.4e+22) 1.0 (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5.4e+22) {
		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 <= 5.4d+22) 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 <= 5.4e+22) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 5.4e+22:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 5.4e+22)
		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 <= 5.4e+22)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 5.4e+22], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.4000000000000004e22

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around inf 47.6%

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

   if 5.4000000000000004e22 < alpha

   1. Initial program 17.4%

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

      1. +-commutative 17.4%

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

   3. Simplified 17.4%

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

   5. Taylor expanded in beta around 0 5.3%

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

      1. +-commutative 5.3%

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

   7. Simplified 5.3%

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

   8. Taylor expanded in alpha around inf 67.2%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.4 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.1% 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 75.5%

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

   1. +-commutative 75.5%

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

3. Simplified 75.5%

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

5. Taylor expanded in beta around inf 38.1%

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

6. Final simplification 38.1%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024093 
(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))