Octave 3.8, jcobi/1

Percentage Accurate: 73.5% → 99.8%

Time: 9.0s

Alternatives: 9

Speedup: 1.3×

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: 73.5% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.95:\\ \;\;\;\;\frac{1 + \left(\beta + 0.5 \cdot \left(\left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta + 2}{\alpha}\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.95)
     (/
      (+
       1.0
       (+ beta (* 0.5 (* (- (- -2.0 beta) beta) (/ (+ beta 2.0) alpha)))))
      alpha)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.95) {
		tmp = (1.0 + (beta + (0.5 * (((-2.0 - beta) - beta) * ((beta + 2.0) / alpha))))) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.95d0)) then
        tmp = (1.0d0 + (beta + (0.5d0 * ((((-2.0d0) - beta) - beta) * ((beta + 2.0d0) / alpha))))) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.95) {
		tmp = (1.0 + (beta + (0.5 * (((-2.0 - beta) - beta) * ((beta + 2.0) / alpha))))) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.95:
		tmp = (1.0 + (beta + (0.5 * (((-2.0 - beta) - beta) * ((beta + 2.0) / alpha))))) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.95)
		tmp = Float64(Float64(1.0 + Float64(beta + Float64(0.5 * Float64(Float64(Float64(-2.0 - beta) - beta) * Float64(Float64(beta + 2.0) / alpha))))) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.95)
		tmp = (1.0 + (beta + (0.5 * (((-2.0 - beta) - beta) * ((beta + 2.0) / alpha))))) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.95], N[(N[(1.0 + N[(beta + N[(0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.94999999999999996

   1. Initial program 8.0%

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

   3. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}\right), \color{blue}{\alpha}\right) \]

   5. Simplified 99.8%

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

   if -0.94999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{1 + \left(\beta + 0.5 \cdot \left(\left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta + 2}{\alpha}\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.0%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \color{blue}{\alpha}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\frac{1}{2} \cdot 2\right) \cdot \beta\right), \alpha\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + 1 \cdot \beta\right), \alpha\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \alpha\right) \]

      8. +-lowering-+.f64 99.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \alpha\right) \]

   5. Simplified 99.1%

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

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

   1. Initial program 99.8%

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

4. Final simplification 99.6%

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

Alternative 3: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 1

   1. Initial program 69.9%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      10. +-lowering-+.f64 69.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 69.9%

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

   5. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\alpha \cdot \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)}\right) \]

   7. Simplified 99.7%

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

   8. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\alpha \cdot \left(1 + 2 \cdot \frac{1}{\alpha}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\alpha \cdot \left(2 \cdot \frac{1}{\alpha} + \color{blue}{1}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(2 \cdot \frac{1}{\alpha}\right) \cdot \alpha + \color{blue}{1 \cdot \alpha}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 \cdot \left(\frac{1}{\alpha} \cdot \alpha\right) + \color{blue}{1} \cdot \alpha\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 \cdot 1 + 1 \cdot \alpha\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 + \color{blue}{1} \cdot \alpha\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 + \alpha\right)\right) \]

      8. +-lowering-+.f64 97.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\alpha}\right)\right) \]

   10. Simplified 97.3%

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

   if 1 < beta

   1. Initial program 86.3%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      10. +-lowering-+.f64 86.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 86.3%

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

   5. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\alpha \cdot \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)}\right) \]

   7. Simplified 61.4%

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

   8. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{\alpha \cdot \left(1 + \frac{1}{\alpha}\right)}{\beta}\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\alpha \cdot \left(1 + \frac{1}{\alpha}\right)}{\beta}\right)}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\alpha \cdot \left(1 + \frac{1}{\alpha}\right)\right), \color{blue}{\beta}\right)\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\alpha \cdot 1 + \alpha \cdot \frac{1}{\alpha}\right), \beta\right)\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\alpha + \alpha \cdot \frac{1}{\alpha}\right), \beta\right)\right)\right) \]

      5. rgt-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right)\right)\right) \]

      6. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right)\right)\right) \]

   10. Simplified 99.9%

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

4. Final simplification 98.3%

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

Alternative 4: 93.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 20:\\ \;\;\;\;\frac{\beta + 1}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 20.0) (/ (+ beta 1.0) (+ beta 2.0)) (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 20.0) {
		tmp = (beta + 1.0) / (beta + 2.0);
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 20.0d0) then
        tmp = (beta + 1.0d0) / (beta + 2.0d0)
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 20.0) {
		tmp = (beta + 1.0) / (beta + 2.0);
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 20.0:
		tmp = (beta + 1.0) / (beta + 2.0)
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 20.0)
		tmp = Float64(Float64(beta + 1.0) / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 20.0)
		tmp = (beta + 1.0) / (beta + 2.0);
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 20.0], N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 20

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      10. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\alpha \cdot \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)}\right) \]

   7. Simplified 85.9%

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

   8. Taylor expanded in alpha around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \color{blue}{\left(2 + \beta\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \beta + 2\right)\right), \left(2 + \beta\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot \beta\right) + \frac{1}{2} \cdot 2\right), \left(\color{blue}{2} + \beta\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot 2\right) \cdot \beta + \frac{1}{2} \cdot 2\right), \left(2 + \beta\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot \beta + \frac{1}{2} \cdot 2\right), \left(2 + \beta\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta + \frac{1}{2} \cdot 2\right), \left(2 + \beta\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta + 1\right), \left(2 + \beta\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\color{blue}{2} + \beta\right)\right) \]

      10. +-lowering-+.f64 99.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(2, \color{blue}{\beta}\right)\right) \]

   10. Simplified 99.0%

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

   if 20 < alpha

   1. Initial program 24.8%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \color{blue}{\alpha}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\frac{1}{2} \cdot 2\right) \cdot \beta\right), \alpha\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + 1 \cdot \beta\right), \alpha\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \alpha\right) \]

      8. +-lowering-+.f64 82.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \alpha\right) \]

   5. Simplified 82.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 20:\\ \;\;\;\;\frac{\beta + 1}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 26.5:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 26.5) (/ 1.0 (+ alpha 2.0)) (+ 1.0 (/ -1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 26.5) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 26.5d0) then
        tmp = 1.0d0 / (alpha + 2.0d0)
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 26.5) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 26.5:
		tmp = 1.0 / (alpha + 2.0)
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 26.5)
		tmp = Float64(1.0 / Float64(alpha + 2.0));
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 26.5)
		tmp = 1.0 / (alpha + 2.0);
	else
		tmp = 1.0 + (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 26.5], N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 26.5

   1. Initial program 69.9%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      10. +-lowering-+.f64 69.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 69.9%

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

   5. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\alpha \cdot \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)}\right) \]

   7. Simplified 99.7%

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

   8. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\alpha \cdot \left(1 + 2 \cdot \frac{1}{\alpha}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\alpha \cdot \left(2 \cdot \frac{1}{\alpha} + \color{blue}{1}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(2 \cdot \frac{1}{\alpha}\right) \cdot \alpha + \color{blue}{1 \cdot \alpha}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 \cdot \left(\frac{1}{\alpha} \cdot \alpha\right) + \color{blue}{1} \cdot \alpha\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 \cdot 1 + 1 \cdot \alpha\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 + \color{blue}{1} \cdot \alpha\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 + \alpha\right)\right) \]

      8. +-lowering-+.f64 97.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\alpha}\right)\right) \]

   10. Simplified 97.3%

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

   if 26.5 < beta

   1. Initial program 86.3%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 84.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 84.3%

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

   6. Taylor expanded in beta around inf

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

      1. sub-neg N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\beta}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\beta}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\beta}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\beta}\right)\right) \]

      5. /-lowering-/.f64 84.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\beta}\right)\right) \]

   8. Simplified 84.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 26.5:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (+ 0.5 (* beta 0.25)) (+ 1.0 (/ -1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * 0.25);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	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 = 0.5d0 + (beta * 0.25d0)
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

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

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5 + (beta * 0.25)
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(beta * 0.25));
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.5 + (beta * 0.25);
	else
		tmp = 1.0 + (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 69.9%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 67.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 67.4%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{4} \cdot \beta} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\beta \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 66.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 66.4%

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

   if 2 < beta

   1. Initial program 86.3%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 84.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 84.3%

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

   6. Taylor expanded in beta around inf

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

      1. sub-neg N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\beta}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\beta}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\beta}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\beta}\right)\right) \]

      5. /-lowering-/.f64 84.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\beta}\right)\right) \]

   8. Simplified 84.3%

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

Alternative 7: 70.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (+ 0.5 (* beta 0.25)) 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * 0.25);
	} 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 = 0.5d0 + (beta * 0.25d0)
    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 = 0.5 + (beta * 0.25);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5 + (beta * 0.25)
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(beta * 0.25));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.5 + (beta * 0.25);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 69.9%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 67.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 67.4%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{4} \cdot \beta} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\beta \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 66.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 66.4%

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

   if 2 < beta

   1. Initial program 86.3%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 84.2%

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

Alternative 8: 70.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 2.1) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.1d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.1)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.1], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 69.9%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 67.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 67.4%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 65.6%

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

   if 2.10000000000000009 < beta

   1. Initial program 86.3%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 84.2%

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

Alternative 9: 49.1% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 75.9%

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

3. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

   2. +-lowering-+.f64 73.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

5. Simplified 73.6%

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

6. Taylor expanded in beta around 0

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

8. Simplified 47.7%

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

Reproduce

herbie shell --seed 2024150 
(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))