Octave 3.8, jcobi/1

Percentage Accurate: 74.3% → 99.6%

Time: 8.7s

Alternatives: 9

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.3% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9:\\ \;\;\;\;\frac{\left(\beta + \left(\beta + 2\right) \cdot \frac{-1 - \beta}{\alpha}\right) - -1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9)
     (/ (- (+ beta (* (+ beta 2.0) (/ (- -1.0 beta) alpha))) -1.0) alpha)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9) {
		tmp = ((beta + ((beta + 2.0) * ((-1.0 - beta) / alpha))) - -1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.9d0)) then
        tmp = ((beta + ((beta + 2.0d0) * (((-1.0d0) - beta) / alpha))) - (-1.0d0)) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9) {
		tmp = ((beta + ((beta + 2.0) * ((-1.0 - beta) / alpha))) - -1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.9:
		tmp = ((beta + ((beta + 2.0) * ((-1.0 - beta) / alpha))) - -1.0) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.9)
		tmp = Float64(Float64(Float64(beta + Float64(Float64(beta + 2.0) * Float64(Float64(-1.0 - beta) / alpha))) - -1.0) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.9)
		tmp = ((beta + ((beta + 2.0) * ((-1.0 - beta) / alpha))) - -1.0) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9], N[(N[(N[(beta + N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(-1.0 - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.900000000000000022

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

      2. sub-neg 6.6%

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

      3. +-commutative 6.6%

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

      4. neg-sub0 6.6%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 6.6%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 6.6%

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

      7. distribute-frac-neg 6.6%

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

      8. +-commutative 6.6%

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

      9. sub-neg 6.6%

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

      10. div-sub 6.6%

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

      11. sub-neg 6.6%

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

      12. metadata-eval 6.6%

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

      13. neg-mul-1 6.6%

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

      14. *-commutative 6.6%

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

      15. +-commutative 6.6%

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

      16. associate-/l/ 6.6%

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

      17. associate-*l/ 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in alpha around inf 96.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right) + 0.5 \cdot \frac{\left(2 + \beta\right) \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right)}{\alpha}}{\alpha}} \]
   6. Step-by-step derivation

      1. fma-define 96.2%

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

      2. neg-mul-1 96.2%

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

      3. +-commutative 96.2%

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

      4. associate-/l* 99.9%

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

      5. +-commutative 99.9%

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

      6. neg-mul-1 99.9%

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

      7. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around -inf 96.2%

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

   10. Simplified 99.9%

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

   if -0.900000000000000022 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9:\\ \;\;\;\;\frac{\left(\beta + \left(\beta + 2\right) \cdot \frac{-1 - \beta}{\alpha}\right) - -1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9) (/ (+ beta 1.0) alpha) (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.9d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.9:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.9)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.9)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.900000000000000022

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

      2. sub-neg 6.6%

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

      3. +-commutative 6.6%

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

      4. neg-sub0 6.6%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 6.6%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 6.6%

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

      7. distribute-frac-neg 6.6%

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

      8. +-commutative 6.6%

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

      9. sub-neg 6.6%

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

      10. div-sub 6.6%

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

      11. sub-neg 6.6%

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

      12. metadata-eval 6.6%

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

      13. neg-mul-1 6.6%

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

      14. *-commutative 6.6%

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

      15. +-commutative 6.6%

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

      16. associate-/l/ 6.6%

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

      17. associate-*l/ 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in alpha around inf 96.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right) + 0.5 \cdot \frac{\left(2 + \beta\right) \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right)}{\alpha}}{\alpha}} \]
   6. Step-by-step derivation

      1. fma-define 96.2%

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

      2. neg-mul-1 96.2%

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

      3. +-commutative 96.2%

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

      4. associate-/l* 99.9%

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

      5. +-commutative 99.9%

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

      6. neg-mul-1 99.9%

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

      7. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. metadata-eval 99.3%

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

      4. associate-*r* 99.3%

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

      5. metadata-eval 99.3%

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

      6. *-lft-identity 99.3%

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

      7. +-commutative 99.3%

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

   10. Simplified 99.3%

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

   if -0.900000000000000022 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{-133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.8:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (* alpha 0.5)) 2.0)))
   (if (<= alpha 2.3e-133)
     t_0
     (if (<= alpha 1.5e-103)
       1.0
       (if (<= alpha 1.8) t_0 (/ (+ beta 1.0) alpha))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= 2.3e-133) {
		tmp = t_0;
	} else if (alpha <= 1.5e-103) {
		tmp = 1.0;
	} else if (alpha <= 1.8) {
		tmp = t_0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    if (alpha <= 2.3d-133) then
        tmp = t_0
    else if (alpha <= 1.5d-103) then
        tmp = 1.0d0
    else if (alpha <= 1.8d0) then
        tmp = t_0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= 2.3e-133) {
		tmp = t_0;
	} else if (alpha <= 1.5e-103) {
		tmp = 1.0;
	} else if (alpha <= 1.8) {
		tmp = t_0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 - (alpha * 0.5)) / 2.0
	tmp = 0
	if alpha <= 2.3e-133:
		tmp = t_0
	elif alpha <= 1.5e-103:
		tmp = 1.0
	elif alpha <= 1.8:
		tmp = t_0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0)
	tmp = 0.0
	if (alpha <= 2.3e-133)
		tmp = t_0;
	elseif (alpha <= 1.5e-103)
		tmp = 1.0;
	elseif (alpha <= 1.8)
		tmp = t_0;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	tmp = 0.0;
	if (alpha <= 2.3e-133)
		tmp = t_0;
	elseif (alpha <= 1.5e-103)
		tmp = 1.0;
	elseif (alpha <= 1.8)
		tmp = t_0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 2.3e-133], t$95$0, If[LessEqual[alpha, 1.5e-103], 1.0, If[LessEqual[alpha, 1.8], t$95$0, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 2.3e-133 or 1.5e-103 < alpha < 1.80000000000000004

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 75.7%

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

   6. Taylor expanded in alpha around 0 75.1%

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

      1. *-commutative 75.1%

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

   8. Simplified 75.1%

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

   if 2.3e-133 < alpha < 1.5e-103

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 77.8%

      \[\leadsto \color{blue}{1} \]

   if 1.80000000000000004 < alpha

   1. Initial program 17.0%

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

      1. +-commutative 17.0%

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

      2. sub-neg 17.0%

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

      3. +-commutative 17.0%

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

      4. neg-sub0 17.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 17.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 17.0%

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

      7. distribute-frac-neg 17.0%

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

      8. +-commutative 17.0%

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

      9. sub-neg 17.0%

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

      10. div-sub 17.0%

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

      11. sub-neg 17.0%

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

      12. metadata-eval 17.0%

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

      13. neg-mul-1 17.0%

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

      14. *-commutative 17.0%

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

      15. +-commutative 17.0%

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

      16. associate-/l/ 17.0%

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

      17. associate-*l/ 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in alpha around inf 85.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right) + 0.5 \cdot \frac{\left(2 + \beta\right) \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right)}{\alpha}}{\alpha}} \]
   6. Step-by-step derivation

      1. fma-define 85.9%

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

      2. neg-mul-1 85.9%

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

      3. +-commutative 85.9%

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

      4. associate-/l* 89.2%

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

      5. +-commutative 89.2%

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

      6. neg-mul-1 89.2%

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

      7. +-commutative 89.2%

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

   7. Simplified 89.2%

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

   8. Taylor expanded in alpha around inf 89.3%

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

      1. associate-*r/ 89.3%

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

      2. distribute-lft-in 89.3%

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

      3. metadata-eval 89.3%

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

      4. associate-*r* 89.3%

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

      5. metadata-eval 89.3%

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

      6. *-lft-identity 89.3%

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

      7. +-commutative 89.3%

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

   10. Simplified 89.3%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{-133}:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.8:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{-133}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 5.6:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.3e-133)
   0.5
   (if (<= alpha 1.5e-103)
     1.0
     (if (<= alpha 5.6) 0.5 (/ (+ beta 1.0) alpha)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.3e-133) {
		tmp = 0.5;
	} else if (alpha <= 1.5e-103) {
		tmp = 1.0;
	} else if (alpha <= 5.6) {
		tmp = 0.5;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.3d-133) then
        tmp = 0.5d0
    else if (alpha <= 1.5d-103) then
        tmp = 1.0d0
    else if (alpha <= 5.6d0) then
        tmp = 0.5d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.3e-133) {
		tmp = 0.5;
	} else if (alpha <= 1.5e-103) {
		tmp = 1.0;
	} else if (alpha <= 5.6) {
		tmp = 0.5;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.3e-133:
		tmp = 0.5
	elif alpha <= 1.5e-103:
		tmp = 1.0
	elif alpha <= 5.6:
		tmp = 0.5
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.3e-133)
		tmp = 0.5;
	elseif (alpha <= 1.5e-103)
		tmp = 1.0;
	elseif (alpha <= 5.6)
		tmp = 0.5;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.3e-133)
		tmp = 0.5;
	elseif (alpha <= 1.5e-103)
		tmp = 1.0;
	elseif (alpha <= 5.6)
		tmp = 0.5;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.3e-133], 0.5, If[LessEqual[alpha, 1.5e-103], 1.0, If[LessEqual[alpha, 5.6], 0.5, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 2.3e-133 or 1.5e-103 < alpha < 5.5999999999999996

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 75.7%

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

   6. Taylor expanded in alpha around 0 73.4%

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

   if 2.3e-133 < alpha < 1.5e-103

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 77.8%

      \[\leadsto \color{blue}{1} \]

   if 5.5999999999999996 < alpha

   1. Initial program 17.0%

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

      1. +-commutative 17.0%

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

      2. sub-neg 17.0%

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

      3. +-commutative 17.0%

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

      4. neg-sub0 17.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 17.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 17.0%

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

      7. distribute-frac-neg 17.0%

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

      8. +-commutative 17.0%

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

      9. sub-neg 17.0%

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

      10. div-sub 17.0%

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

      11. sub-neg 17.0%

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

      12. metadata-eval 17.0%

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

      13. neg-mul-1 17.0%

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

      14. *-commutative 17.0%

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

      15. +-commutative 17.0%

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

      16. associate-/l/ 17.0%

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

      17. associate-*l/ 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in alpha around inf 85.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right) + 0.5 \cdot \frac{\left(2 + \beta\right) \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right)}{\alpha}}{\alpha}} \]
   6. Step-by-step derivation

      1. fma-define 85.9%

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

      2. neg-mul-1 85.9%

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

      3. +-commutative 85.9%

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

      4. associate-/l* 89.2%

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

      5. +-commutative 89.2%

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

      6. neg-mul-1 89.2%

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

      7. +-commutative 89.2%

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

   7. Simplified 89.2%

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

   8. Taylor expanded in alpha around inf 89.3%

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

      1. associate-*r/ 89.3%

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

      2. distribute-lft-in 89.3%

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

      3. metadata-eval 89.3%

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

      4. associate-*r* 89.3%

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

      5. metadata-eval 89.3%

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

      6. *-lft-identity 89.3%

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

      7. +-commutative 89.3%

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

   10. Simplified 89.3%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{-133}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 5.6:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{-133}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.8 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.9:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.3e-133)
   0.5
   (if (<= alpha 1.8e-103) 1.0 (if (<= alpha 0.9) 0.5 (/ 1.0 alpha)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.3e-133) {
		tmp = 0.5;
	} else if (alpha <= 1.8e-103) {
		tmp = 1.0;
	} else if (alpha <= 0.9) {
		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 <= 2.3d-133) then
        tmp = 0.5d0
    else if (alpha <= 1.8d-103) then
        tmp = 1.0d0
    else if (alpha <= 0.9d0) 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 <= 2.3e-133) {
		tmp = 0.5;
	} else if (alpha <= 1.8e-103) {
		tmp = 1.0;
	} else if (alpha <= 0.9) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.3e-133:
		tmp = 0.5
	elif alpha <= 1.8e-103:
		tmp = 1.0
	elif alpha <= 0.9:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.3e-133)
		tmp = 0.5;
	elseif (alpha <= 1.8e-103)
		tmp = 1.0;
	elseif (alpha <= 0.9)
		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 <= 2.3e-133)
		tmp = 0.5;
	elseif (alpha <= 1.8e-103)
		tmp = 1.0;
	elseif (alpha <= 0.9)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.3e-133], 0.5, If[LessEqual[alpha, 1.8e-103], 1.0, If[LessEqual[alpha, 0.9], 0.5, N[(1.0 / alpha), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 2.3e-133 or 1.7999999999999999e-103 < alpha < 0.900000000000000022

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 75.7%

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

   6. Taylor expanded in alpha around 0 73.4%

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

   if 2.3e-133 < alpha < 1.7999999999999999e-103

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 77.8%

      \[\leadsto \color{blue}{1} \]

   if 0.900000000000000022 < alpha

   1. Initial program 17.0%

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

      1. +-commutative 17.0%

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

      2. sub-neg 17.0%

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

      3. +-commutative 17.0%

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

      4. neg-sub0 17.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 17.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 17.0%

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

      7. distribute-frac-neg 17.0%

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

      8. +-commutative 17.0%

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

      9. sub-neg 17.0%

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

      10. div-sub 17.0%

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

      11. sub-neg 17.0%

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

      12. metadata-eval 17.0%

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

      13. neg-mul-1 17.0%

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

      14. *-commutative 17.0%

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

      15. +-commutative 17.0%

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

      16. associate-/l/ 17.0%

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

      17. associate-*l/ 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in alpha around inf 85.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right) + 0.5 \cdot \frac{\left(2 + \beta\right) \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right)}{\alpha}}{\alpha}} \]
   6. Step-by-step derivation

      1. fma-define 85.9%

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

      2. neg-mul-1 85.9%

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

      3. +-commutative 85.9%

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

      4. associate-/l* 89.2%

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

      5. +-commutative 89.2%

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

      6. neg-mul-1 89.2%

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

      7. +-commutative 89.2%

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

   7. Simplified 89.2%

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

   8. Taylor expanded in beta around 0 74.6%

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

      1. associate-*r/ 74.6%

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

      2. metadata-eval 74.6%

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

   10. Simplified 74.6%

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

   11. Taylor expanded in alpha around inf 74.5%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{-133}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.8 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.9:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 330000000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 330000000000.0)
   (+ 0.5 (* (- alpha beta) (/ -0.5 (+ beta (+ alpha 2.0)))))
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 330000000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 330000000000.0d0) then
        tmp = 0.5d0 + ((alpha - beta) * ((-0.5d0) / (beta + (alpha + 2.0d0))))
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 330000000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 330000000000.0:
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))))
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 330000000000.0)
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) * Float64(-0.5 / Float64(beta + Float64(alpha + 2.0)))));
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 330000000000.0)
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 330000000000.0], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.3e11

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   if 3.3e11 < alpha

   1. Initial program 16.0%

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

      1. +-commutative 16.0%

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

      2. sub-neg 16.0%

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

      3. +-commutative 16.0%

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

      4. neg-sub0 16.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 16.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 16.0%

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

      7. distribute-frac-neg 16.0%

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

      8. +-commutative 16.0%

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

      9. sub-neg 16.0%

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

      10. div-sub 16.0%

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

      11. sub-neg 16.0%

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

      12. metadata-eval 16.0%

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

      13. neg-mul-1 16.0%

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

      14. *-commutative 16.0%

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

      15. +-commutative 16.0%

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

      16. associate-/l/ 16.0%

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

      17. associate-*l/ 16.0%

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

   3. Simplified 16.1%

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

   5. Taylor expanded in alpha around inf 86.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right) + 0.5 \cdot \frac{\left(2 + \beta\right) \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right)}{\alpha}}{\alpha}} \]
   6. Step-by-step derivation

      1. fma-define 86.6%

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

      2. neg-mul-1 86.6%

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

      3. +-commutative 86.6%

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

      4. associate-/l* 90.0%

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

      5. +-commutative 90.0%

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

      6. neg-mul-1 90.0%

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

      7. +-commutative 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in alpha around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. distribute-lft-in 90.0%

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

      3. metadata-eval 90.0%

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

      4. associate-*r* 90.0%

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

      5. metadata-eval 90.0%

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

      6. *-lft-identity 90.0%

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

      7. +-commutative 90.0%

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

   10. Simplified 90.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 330000000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 13.6:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 13.6)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 13.6) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 13.6d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 13.6) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 13.6:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 13.6)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 13.6)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 13.6], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 13.5999999999999996

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. +-commutative 97.9%

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

   7. Simplified 97.9%

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

   if 13.5999999999999996 < alpha

   1. Initial program 17.0%

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

      1. +-commutative 17.0%

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

      2. sub-neg 17.0%

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

      3. +-commutative 17.0%

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

      4. neg-sub0 17.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 17.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 17.0%

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

      7. distribute-frac-neg 17.0%

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

      8. +-commutative 17.0%

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

      9. sub-neg 17.0%

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

      10. div-sub 17.0%

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

      11. sub-neg 17.0%

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

      12. metadata-eval 17.0%

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

      13. neg-mul-1 17.0%

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

      14. *-commutative 17.0%

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

      15. +-commutative 17.0%

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

      16. associate-/l/ 17.0%

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

      17. associate-*l/ 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in alpha around inf 85.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right) + 0.5 \cdot \frac{\left(2 + \beta\right) \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right)}{\alpha}}{\alpha}} \]
   6. Step-by-step derivation

      1. fma-define 85.9%

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

      2. neg-mul-1 85.9%

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

      3. +-commutative 85.9%

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

      4. associate-/l* 89.2%

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

      5. +-commutative 89.2%

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

      6. neg-mul-1 89.2%

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

      7. +-commutative 89.2%

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

   7. Simplified 89.2%

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

   8. Taylor expanded in alpha around inf 89.3%

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

      1. associate-*r/ 89.3%

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

      2. distribute-lft-in 89.3%

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

      3. metadata-eval 89.3%

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

      4. associate-*r* 89.3%

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

      5. metadata-eval 89.3%

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

      6. *-lft-identity 89.3%

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

      7. +-commutative 89.3%

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

   10. Simplified 89.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 13.6:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 12.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		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 <= 12.0d0) 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 <= 12.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 12.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 12.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 12.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 12.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 12

   1. Initial program 65.4%

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

      1. +-commutative 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in beta around 0 63.6%

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

   6. Taylor expanded in alpha around 0 61.0%

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

   if 12 < beta

   1. Initial program 84.3%

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

      1. +-commutative 84.3%

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

      2. sub-neg 84.3%

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

      3. +-commutative 84.3%

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

      4. neg-sub0 84.3%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 84.3%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 84.3%

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

      7. distribute-frac-neg 84.3%

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

      8. +-commutative 84.3%

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

      9. sub-neg 84.3%

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

      10. div-sub 84.3%

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

      11. sub-neg 84.3%

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

      12. metadata-eval 84.3%

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

      13. neg-mul-1 84.3%

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

      14. *-commutative 84.3%

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

      15. +-commutative 84.3%

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

      16. associate-/l/ 84.3%

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

      17. associate-*l/ 84.3%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in beta around inf 81.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.8% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 70.8%

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

   1. +-commutative 70.8%

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

   2. sub-neg 70.8%

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

   3. +-commutative 70.8%

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

   4. neg-sub0 70.8%

      \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

   5. associate-+l- 70.8%

      \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

   6. sub0-neg 70.8%

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

   7. distribute-frac-neg 70.8%

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

   8. +-commutative 70.8%

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

   9. sub-neg 70.8%

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

   10. div-sub 70.8%

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

   11. sub-neg 70.8%

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

   12. metadata-eval 70.8%

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

   13. neg-mul-1 70.8%

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

   14. *-commutative 70.8%

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

   15. +-commutative 70.8%

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

   16. associate-/l/ 70.8%

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

   17. associate-*l/ 70.8%

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

3. Simplified 70.8%

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

5. Taylor expanded in beta around inf 33.0%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 33.0%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(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))