Octave 3.8, jcobi/1

Percentage Accurate: 75.3% → 99.7%

Time: 9.5s

Alternatives: 11

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: 75.3% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \beta \cdot 2\\ \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\alpha \cdot \left(\frac{-2}{\alpha} \cdot \frac{\left(\beta + 2\right) \cdot \left(\left(-2 - \beta\right) - \beta\right)}{t\_0 \cdot t\_0} + \frac{2}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1 + \alpha}{\beta}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* beta 2.0))))
   (if (<= beta 3.5e+16)
     (/
      1.0
      (*
       alpha
       (+
        (*
         (/ -2.0 alpha)
         (/ (* (+ beta 2.0) (- (- -2.0 beta) beta)) (* t_0 t_0)))
        (/ 2.0 t_0))))
     (/ 1.0 (+ 1.0 (/ (+ 1.0 alpha) beta))))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta * 2.0);
	double tmp;
	if (beta <= 3.5e+16) {
		tmp = 1.0 / (alpha * (((-2.0 / alpha) * (((beta + 2.0) * ((-2.0 - beta) - beta)) / (t_0 * t_0))) + (2.0 / t_0)));
	} else {
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (beta * 2.0d0)
    if (beta <= 3.5d+16) then
        tmp = 1.0d0 / (alpha * ((((-2.0d0) / alpha) * (((beta + 2.0d0) * (((-2.0d0) - beta) - beta)) / (t_0 * t_0))) + (2.0d0 / t_0)))
    else
        tmp = 1.0d0 / (1.0d0 + ((1.0d0 + alpha) / beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta * 2.0);
	double tmp;
	if (beta <= 3.5e+16) {
		tmp = 1.0 / (alpha * (((-2.0 / alpha) * (((beta + 2.0) * ((-2.0 - beta) - beta)) / (t_0 * t_0))) + (2.0 / t_0)));
	} else {
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 2.0 + (beta * 2.0)
	tmp = 0
	if beta <= 3.5e+16:
		tmp = 1.0 / (alpha * (((-2.0 / alpha) * (((beta + 2.0) * ((-2.0 - beta) - beta)) / (t_0 * t_0))) + (2.0 / t_0)))
	else:
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta * 2.0))
	tmp = 0.0
	if (beta <= 3.5e+16)
		tmp = Float64(1.0 / Float64(alpha * Float64(Float64(Float64(-2.0 / alpha) * Float64(Float64(Float64(beta + 2.0) * Float64(Float64(-2.0 - beta) - beta)) / Float64(t_0 * t_0))) + Float64(2.0 / t_0))));
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(Float64(1.0 + alpha) / beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta * 2.0);
	tmp = 0.0;
	if (beta <= 3.5e+16)
		tmp = 1.0 / (alpha * (((-2.0 / alpha) * (((beta + 2.0) * ((-2.0 - beta) - beta)) / (t_0 * t_0))) + (2.0 / t_0)));
	else
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.5e+16], N[(1.0 / N[(alpha * N[(N[(N[(-2.0 / alpha), $MachinePrecision] * N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5e16

   1. Initial program 72.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 72.8%

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

   4. Applied egg-rr 72.8%

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

   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%

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

   if 3.5e16 < beta

   1. Initial program 83.4%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 83.4%

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

   4. Applied egg-rr 83.4%

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

   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 49.2%

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

   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. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. rgt-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. +-lowering-+.f64 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\alpha \cdot \left(\frac{-2}{\alpha} \cdot \frac{\left(\beta + 2\right) \cdot \left(\left(-2 - \beta\right) - \beta\right)}{\left(2 + \beta \cdot 2\right) \cdot \left(2 + \beta \cdot 2\right)} + \frac{2}{2 + \beta \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1 + \alpha}{\beta}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 5.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 100.0%

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

   5. Simplified 100.0%

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

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

   1. Initial program 99.3%

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

      1. div-sub N/A

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

      2. associate-+l- N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

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

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

      11. /-lowering-/.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

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

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

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

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

      16. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 99.5%

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

Alternative 3: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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 100.0%

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

   5. Simplified 100.0%

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

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

   1. Initial program 99.3%

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

4. Final simplification 99.5%

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

Alternative 4: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 44:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1 + \alpha}{\beta}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.4e-9)
   (/ 1.0 (+ alpha 2.0))
   (if (<= beta 44.0)
     (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
     (/ 1.0 (+ 1.0 (/ (+ 1.0 alpha) beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4e-9) {
		tmp = 1.0 / (alpha + 2.0);
	} else if (beta <= 44.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / 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 <= 6.4d-9) then
        tmp = 1.0d0 / (alpha + 2.0d0)
    else if (beta <= 44.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = 1.0d0 / (1.0d0 + ((1.0d0 + alpha) / beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4e-9) {
		tmp = 1.0 / (alpha + 2.0);
	} else if (beta <= 44.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.4e-9:
		tmp = 1.0 / (alpha + 2.0)
	elif beta <= 44.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.4e-9)
		tmp = Float64(1.0 / Float64(alpha + 2.0));
	elseif (beta <= 44.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(Float64(1.0 + alpha) / beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.4e-9)
		tmp = 1.0 / (alpha + 2.0);
	elseif (beta <= 44.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.4e-9], N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 44.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(1.0 + N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.40000000000000023e-9

   1. Initial program 72.2%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 72.2%

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

   4. Applied egg-rr 72.2%

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

   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 99.8%

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

   8. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-lowering-+.f64 99.1%

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

   10. Simplified 99.1%

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

   if 6.40000000000000023e-9 < beta < 44

   1. Initial program 89.3%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 90.0%

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

   5. Simplified 90.0%

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

   if 44 < beta

   1. Initial program 82.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 82.5%

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

   4. Applied egg-rr 82.5%

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

   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 50.7%

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

   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. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. rgt-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. +-lowering-+.f64 98.8%

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

   10. Simplified 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 44:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1 + \alpha}{\beta}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 44:\\ \;\;\;\;\frac{\beta + 1}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1 + \alpha}{\beta}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.6e-9)
   (/ 1.0 (+ alpha 2.0))
   (if (<= beta 44.0)
     (/ (+ beta 1.0) (+ beta 2.0))
     (/ 1.0 (+ 1.0 (/ (+ 1.0 alpha) beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6e-9) {
		tmp = 1.0 / (alpha + 2.0);
	} else if (beta <= 44.0) {
		tmp = (beta + 1.0) / (beta + 2.0);
	} else {
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / 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 <= 6.6d-9) then
        tmp = 1.0d0 / (alpha + 2.0d0)
    else if (beta <= 44.0d0) then
        tmp = (beta + 1.0d0) / (beta + 2.0d0)
    else
        tmp = 1.0d0 / (1.0d0 + ((1.0d0 + alpha) / beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6e-9) {
		tmp = 1.0 / (alpha + 2.0);
	} else if (beta <= 44.0) {
		tmp = (beta + 1.0) / (beta + 2.0);
	} else {
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.6e-9:
		tmp = 1.0 / (alpha + 2.0)
	elif beta <= 44.0:
		tmp = (beta + 1.0) / (beta + 2.0)
	else:
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.6e-9)
		tmp = Float64(1.0 / Float64(alpha + 2.0));
	elseif (beta <= 44.0)
		tmp = Float64(Float64(beta + 1.0) / Float64(beta + 2.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(Float64(1.0 + alpha) / beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.6e-9)
		tmp = 1.0 / (alpha + 2.0);
	elseif (beta <= 44.0)
		tmp = (beta + 1.0) / (beta + 2.0);
	else
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.6e-9], N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 44.0], N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.60000000000000037e-9

   1. Initial program 72.2%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 72.2%

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

   4. Applied egg-rr 72.2%

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

   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 99.8%

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

   8. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-lowering-+.f64 99.1%

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

   10. Simplified 99.1%

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

   if 6.60000000000000037e-9 < beta < 44

   1. Initial program 89.3%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 88.9%

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

   4. Applied egg-rr 88.9%

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

   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 99.0%

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

   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. +-commutative N/A

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

      11. +-lowering-+.f64 89.6%

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

   10. Simplified 89.6%

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

   if 44 < beta

   1. Initial program 82.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 82.5%

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

   4. Applied egg-rr 82.5%

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

   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 50.7%

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

   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. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. rgt-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. +-lowering-+.f64 98.8%

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

   10. Simplified 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 44:\\ \;\;\;\;\frac{\beta + 1}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1 + \alpha}{\beta}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.45 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\beta + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.45e-9) (/ 1.0 (+ alpha 2.0)) (/ (+ beta 1.0) (+ beta 2.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.45e-9) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = (beta + 1.0) / (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 <= 2.45d-9) then
        tmp = 1.0d0 / (alpha + 2.0d0)
    else
        tmp = (beta + 1.0d0) / (beta + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.45e-9) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = (beta + 1.0) / (beta + 2.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.45e-9:
		tmp = 1.0 / (alpha + 2.0)
	else:
		tmp = (beta + 1.0) / (beta + 2.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.45e-9)
		tmp = Float64(1.0 / Float64(alpha + 2.0));
	else
		tmp = Float64(Float64(beta + 1.0) / Float64(beta + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.45e-9)
		tmp = 1.0 / (alpha + 2.0);
	else
		tmp = (beta + 1.0) / (beta + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.45e-9], N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.45000000000000002e-9

   1. Initial program 72.2%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 72.2%

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

   4. Applied egg-rr 72.2%

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

   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 99.8%

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

   8. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-lowering-+.f64 99.1%

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

   10. Simplified 99.1%

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

   if 2.45000000000000002e-9 < beta

   1. Initial program 83.3%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 83.3%

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

   4. Applied egg-rr 83.3%

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

   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 56.4%

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

   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. +-commutative N/A

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

      11. +-lowering-+.f64 81.8%

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

   10. Simplified 81.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.45 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\beta + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.0) (/ 1.0 (+ alpha 2.0)) (+ 1.0 (/ -1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		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 <= 9.0d0) 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 <= 9.0) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 9.0:
		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 <= 9.0)
		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 <= 9.0)
		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, 9.0], 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 < 9

   1. Initial program 72.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 72.5%

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

   4. Applied egg-rr 72.5%

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

   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 99.8%

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

   8. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-lowering-+.f64 97.4%

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

   10. Simplified 97.4%

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

   if 9 < beta

   1. Initial program 83.3%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 81.6%

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

   5. Simplified 81.6%

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

   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 78.6%

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

   8. Simplified 78.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.96:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.96) (+ 0.5 (* alpha -0.25)) (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.96) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 0.96d0) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.96) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.96:
		tmp = 0.5 + (alpha * -0.25)
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.96)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.96)
		tmp = 0.5 + (alpha * -0.25);
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.96], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.95999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 78.8%

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

   5. Simplified 78.8%

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

   6. Taylor expanded in alpha around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 77.6%

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

   8. Simplified 77.6%

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

   if 0.95999999999999996 < alpha

   1. Initial program 27.9%

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

   3. Taylor expanded in beta around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 7.2%

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

   5. Simplified 7.2%

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

   6. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 64.3%

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

   8. Simplified 64.3%

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

Alternative 9: 68.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 2

   1. Initial program 100.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 77.1%

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

   if 2 < alpha

   1. Initial program 27.9%

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

   3. Taylor expanded in beta around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 7.2%

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

   5. Simplified 7.2%

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

   6. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 64.3%

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

   8. Simplified 64.3%

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

Alternative 10: 71.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 72.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 70.0%

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

   5. Simplified 70.0%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 67.8%

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

   if 2 < beta

   1. Initial program 83.3%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 78.2%

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

Alternative 11: 49.7% 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 75.5%

   \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
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 73.2%

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

5. Simplified 73.2%

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

6. Taylor expanded in beta around 0

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

8. Simplified 53.7%

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

Reproduce

herbie shell --seed 2024161 
(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))